V. Algebraical Functions

1. We shall now consider the integrals of algebraical functions, explicit or implicit. The theory of the integration of such functions is far more extensive and difficult than that of rational functions, and we can give here only a brief account of a few of the most important results and of the most obvious of their applications.

If are algebraical functions of , then any algebraical function of , is an algebraical function of . This is obvious if we confine ourselves to explicit algebraical functions. In the general case we have a number of equations of the type and where the 's represent polynomials in their arguments. The elimination of between these equations gives an equation in whose coefficients are polynomials in only.

The importance of this from our present point of view lies in the fact that we may consider the standard algebraical integral under any of the forms where ; where and is rational; or where . It is, for example, much more convenient to treat such an irrational as as a rational function of , , where , , than as a rational function of and , where To treat it as a simple irrational , so that our fundamental equation is is evidently the least convenient course of all.

Before we proceed to consider the general form of the integral of an algebraical function we shall consider one most important case in which the integral can be at once reduced to that of a rational function, and is therefore always an elementary function itself.

2. The class of integrals alluded to immediately above is that covered by the following theorem.

If there is a variable connected with and (or ) by rational relations (or ), then the integral (or ) is an elementary function.

The truth of this proposition follows immediately from the equations where all the capital letters denote rational functions.

The most important case of this theorem is that in which and are connected by the general quadratic relation The integral can then be made rational in an infinite number of ways. For suppose that is any point on the conic, and that is any line through the point. If we eliminate between these equations, we obtain an equation of the second degree in , say where are polynomials in . But one root of this equation must be , which is independent of ; and when we divide by we obtain an equation of the first degree for the abscissa of the variable point of intersection, in which the coefficients are again polynomials in . Hence this abscissa is a rational function of ; the ordinate of the point is also a rational function of , and as varies this point coincides with every point of the conic in turn. In fact the equation of the conic may be written in the form where , and the other point of intersection of the line and the conic is given by

An alternative method is to write so that and are parallel to the asymptotes of the conic, and to put Then and from these two equations we can calculate and as rational functions of . The principle of this method is of course the same as that of the former method: is now at infinity, and the pencil of lines through is replaced by a pencil parallel to an asymptote.

The most important case is that in which , , so that The integral is then made rational by the substitution where , are any numbers such that We may for instance suppose that , ; or that , while is a root of the equation . Or again the integral is made rational by putting , when

3. We shall now consider in more detail the problem of the calculation of where[17] The most interesting case is that in which , , and the constants which occur in are real, and we shall confine our attention to this case.

Let where and are polynomials. Then, by means of the equation may be reduced to the form where , , , are polynomials in ; and so to the form , where and are rational, or (what is the same thing) the form where and are rational. The rational part may be integrated by the methods of section iv., and the integral may be reduced to the sum of a number of integrals of the forms where , , , , , are real constants and a positive integer. The result is generally required in an explicitly real form: and, as further progress depends on transformations involving (or , , ), it is generally not advisable to break up a quadratic factor into its constituent linear factors when these factors are complex.

All of the integrals (1) may be reduced, by means of elementary formulae of reduction[18], to dependence upon three fundamental integrals, viz.

4. The first of these integrals may be reduced, by a substitution of the type , to one or other of the three standard forms where . These integrals may be rationalised by the substitutions but it is simpler to use the transcendental substitutions These last substitutions are generally the most convenient for the reduction of an integral which contains one or other of the irrationalities though the alternative substitutions are often useful.

It has been pointed out by Dr Bromwich that the forms usually given in text-books for these three standard integrals, viz. are not quite accurate. It is obvious, for example, that the first two of these functions are odd functions of , while the corresponding integrals are even functions. The correct formulae are and where the ambiguous sign is the same as that of . It is in some ways more convenient to use the equivalent forms

5. The integral may be evaluated in a variety of ways.

If is a root of the equation , then may be written in the form , and the value of the integral is given by one or other of the formulae We may therefore suppose that is not a root of .

(i) We may follow the general method described above, taking[19] Eliminating from the equations and dividing by , we obtain and so

Hence But and so If the transformation is imaginary.

Suppose, e.g., (a) , , or (b) , . We find

(a) where or and

(b) where Neither of these results is expressed in the simplest form, the second in particular being very inconvenient.

(ii) The most straightforward method of procedure is to use the substitution We then obtain where are certain simple functions of , and . The further reduction of this integral has been discussed already.

(iii) A third method of integration is that adopted by Sir G. Greenhill[20], who uses the transformation It will be found that which is of one of the three standard forms mentioned in §4.

6. It remains to consider the integral where or is a quadratic with complex linear factors. Here again there is a choice of methods at our disposal.

We may suppose that is not a constant multiple of . If it is, then the value of the integral is given by the formula[21]

(i) The standard method is to use the substitution where and are so chosen that

The values of and which satisfy these conditions are the roots of the quadratic The roots will be real and distinct if or if Now , so that (3) is certainly satisfied if . But if and are both positive then and have the same sign, and Thus the values of and are in any case real and distinct.

It will be found, on carrying out the substitution (1), that where , , , , , and are constants. Of these two integrals, the first is rationalised by the substitution and the second by the substitution[22]

It should be observed that this method fails in the special case in which . In this case, however, the substitution reduces the integral to one of the form and the reduction may then be completed as before.

(ii) An alternative method is to use Sir G. Greenhill's substitution If then The maximum and minimum values of are given by .

Again and the numerator will be a perfect square if

It will be found by a little calculation that the discriminant of this quadratic and that of differ from one another and from where , are the roots of and , those of , only by a constant factor which is always negative. Since and are conjugate complex numbers, this product is positive, and so and have real roots[23]. We denote the roots of the latter by Then say. Further, since can vanish for two equal values of only if is equal to or , i.e. when is a maximum or a minimum, can differ from only by a constant factor; and by comparing coefficients and using the identity we find that

Finally, we can write in the form

Using equations (1), (2), (2'), and (3), we find that and the integral is reduced to a sum of two standard forms.

This method is very elegant, and has the advantage that the whole work of transformation is performed in one step. On the other hand it is somewhat artificial, and it is open to the logical objection that it introduces the root , which, in virtue of Laplace's principle (iii., 2), cannot really be involved in the final result[24].

7. We may now proceed to consider the general case to which the theorem of iv., §2 applies. It will be convenient to recall two well-known definitions in the theory of algebraical plane curves. A curve of degree can have at most double points[25]. If the actual number of double points is , then the number is called the deficiency[26] of the curve.

If the coordinates , of the points on a curve can be expressed rationally in terms of a parameter by means of equations then we shall say that the curve is unicursal. In this case we have seen that we can always evaluate in terms of elementary functions.

The fundamental theorem in this part of our subject is

'A curve whose deficiency is zero is unicursal, and vice versa'.

Suppose first that the curve possesses the maximum number of double points[27]. Since and points are just sufficient to determine a curve of degree [28], we can draw, through the double points and other points chosen arbitrarily on the curve, a simply infinite set of curves of degree , which we may suppose to have the equation where is a variable parameter and , are the equations of two particular members of the set. Any one of these curves meets the given curve in points, of which are accounted for by the double points, and by the other arbitrarily chosen points. These points are independent of ; and so there is but one point of intersection which depends on . The coordinates of this point are given by The elimination of gives an equation of degree in , whose coefficients are polynomials in ; and but one root of this equation varies with . The eliminant is therefore divisible by a factor of degree which does not contain . There remains a simple equation in whose coefficients are polynomials in . Thus the -coordinate of the variable point is determined as a rational function of , and the -coordinate may be similarly determined.

We may therefore write If we reduce these fractions to the same denominator, we express the coordinates in the form where , , are polynomials which have no common factor. The polynomials will in general be of degree ; none of them can be of higher degree, and one at least must be actually of that degree, since an arbitrary straight line must cut the curve in exactly points[29].

We can now prove the second part of the theorem. If where , , are polynomials of degree , then the line will meet the curve in points whose parameters are given by

This equation will have a double root if Hence the equation of the tangent at the point is

If is a fixed point, then the equation (2) may be regarded as an equation to determine the parameters of the points of contact of the tangents from . Now is of degree in , the coefficient of obviously vanishing. Hence in general the number of tangents which can be drawn to a unicursal curve from a fixed point (the class of the curve) is . But the class of a curve whose only singular points are nodes is known[30] to be . Hence the number of nodes is

It is perhaps worth pointing out how the proof which precedes requires modification if some only of the singular points are nodes and the rest ordinary cusps. The first part of the proof remains unaltered. The equation (2) must now be regarded as giving the values of which correspond to (a) points at which the tangent passes through and (b) cusps, since any line through a cusp 'cuts the curve in two coincident points'[31]. We have therefore where is the class of the curve. But[32] and so[33]

8. (i) The preceding argument fails if , but we have already seen that all conics are unicursal. The case next in importance is that of a cubic with a double point. If the double point is not at infinity we can, by a change of origin, reduce the equation of the curve to the form and, by considering the intersections of the curve with the line , we find

If the double point is at infinity, the equation of the curve is of the form the curve having a pair of parallel asymptotes; and, by considering the intersection of the curve with the line , we find

(ii) The case next in complexity is that of a quartic with three double points.

(a) The lemniscate has three double points, the origin and the circular points at infinity. The circle passes through these points and one other fixed point at the origin, as it touches the curve there. Solving, we find

(b) The curve has the double points , , . Using the auxiliary conic we find

(iii) (a) The curve has a multiple point of order at the origin, and is therefore unicursal. In this case it is sufficient to consider the intersection of the curve with the line . This may be harmonised with the general theory by regarding the curve as passing through each of the double points collected at the origin and through other fixed points collected at the point

The curves are projectively equivalent, as appears on rendering their equations homogeneous by the introduction of variables in (1) and in (2). We conclude that (2) is unicursal, having the maximum number of double points at infinity. In fact we may put The integral is accordingly an elementary function.

(b) The curve is unicursal if and only if either (i) or (ii) or (iii) . Hence the integral is an elementary function, for all forms of , in these three cases only; of course it is integrable for special forms of in other cases[34].

9. There is a similar theory connected with unicursal curves in space of any number of dimensions. Consider for example the integral A linear substitution reduces this integral to the form and this integral can be rationalised by putting

The curve whose Cartesian coordinates , , are given by is a unicursal twisted quartic, the intersection of the parabolic cylinders

It is easy to deduce that the integral is always an elementary function.

10. When the deficiency of the curve is not zero, the integral is in general not an elementary function; and the consideration of such integrals has consequently introduced a whole series of classes of new transcendents into analysis. The simplest case is that in which the deficiency is unity: in this case, as we shall see later on, the integrals are expressible in terms of elementary functions and certain new transcendents known as elliptic integrals. When the deficiency rises above unity the integration necessitates the introduction of new transcendents of growing complexity.

But there are infinitely many particular cases in which integrals, associated with curves whose deficiency is unity or greater than unity, can be expressed in terms of elementary functions, or are even algebraical themselves. For instance the deficiency of is unity. But And, before we say anything concerning the new transcendents to which integrals of this class in general give rise, we shall consider what has been done in the way of formulating rules to enable us to identify such cases and to assign the form of the integral when it is an elementary function. It will be as well to say at once that this problem has not been solved completely.

11. The first general theorem of this character deals with the case in which the integral is algebraical, and asserts that if is an algebraical function of , then it is a rational function of and .

Our proof will be based on the following lemmas.

(1) If and are polynomials, and there is no factor common to all the coefficients of the various powers of in ; and where is a rational function of ; then is a polynomial.

Let , where and are polynomials without a common factor. Then If is a factor of , then for all values of ; and so all the coefficients of powers of in are divisible by , which is contrary to our hypotheses. Hence is a constant and a polynomial.

(2) Suppose that is an irreducible polynomial, and that are the roots of in a certain domain . Suppose further that is another polynomial, and that Then where is any one of the roots of (1); and where also is a polynomial in and .

Let us determine the highest common factor of and , considered as polynomials in , by the ordinary process for the determination of the highest common factor of two polynomials. This process depends only on a series of algebraical divisions, and so is a polynomial in with coefficients rational in . We have therefore where , , , , and are polynomials and , , and rational functions; and evidently we may suppose that neither in nor in have the coefficients of all powers of a common factor. Hence, by Lemma (1), and are polynomials. But is irreducible, and therefore and either or must be constants. If were a constant, would be a function of only. But this is impossible. For we can determine polynomials , in , with coefficients rational in , such that and the left-hand side of (4) vanishes when we write for . Hence is a constant, and so is a constant multiple of . The truth of the lemma now follows from (3).

It follows from Lemma (2) that cannot satisfy any equation of degree less than whose coefficients are polynomials in .

(3) If is an algebraical function of , defined by an equation of degree , then any rational function of and can be expressed in the form where are rational functions of .

The function is one of the roots of (1). Let be the complete system of roots. Then where and are polynomials. The denominator is a polynomial in whose coefficients are symmetric polynomials in , and is therefore, by ii., §3, (i), a rational function of . On the other hand is a polynomial in whose coefficients are symmetric polynomials in , and therefore, by ii., §3, (ii), polynomials in with coefficients rational in . Thus the numerator of (3) is a polynomial in with coefficients rational in .

It follows that is a polynomial in with coefficients rational in . From this polynomial we can eliminate, by means of (1), all powers of as high as or higher than the th. Hence is of the form prescribed by the lemma.

12. We proceed now to the proof of our main theorem. We have where is algebraical. Let be the irreducible equations satisfied by and , and let us suppose that they are of degrees and respectively. The first stage in the proof consists in showing that It will be convenient now to write , for , , and to denote by the complete systems of roots of the equations (1).

We have and so Now let Then is a polynomial in , with coefficients symmetric in and therefore rational in .

The equations and have a root in common, and the first equation is irreducible. It follows, by Lemma (2) of §11, that for .[35] And from this it follows that, when is given, we have for some value of the suffix .

But we have also and from (2) and (3) it follows[36] that i.e. that every is the integral of some .

In the same way we can show that every is the derivative of some . Let Then is a polynomial in , with coefficients symmetric in and therefore rational in . The equations and have a root in common, and so for . From this we deduce that, when is given, (2) must be true for some value of , and so that the same is true of (4).

Now it is impossible that, in (4), two different values of should correspond to the same value of . For this would involve where and is a constant. Hence we should have Subtracting these equations, we should obtain an equation of degree in , with coefficients which are polynomials in ; and this is impossible. In the same way we can prove that two different values of cannot correspond to the same value of .

The equation (4) therefore establishes a one-one correspondence between the values of and . It follows that It is moreover evident that, by arranging the suffixes properly, we can make for .

13. We have where is a rational function which may, in virtue of Lemma (3) of § 11, be expressed as a polynomial of degree in , with coefficients rational in .

The product is a polynomial of degree in , with coefficients which are symmetric polynomials in and therefore, by ii., §3, (ii), polynomials in with coefficients rational in . Replacing by its expression as a polynomial in obtained above, and eliminating and all higher powers of , we obtain an equation where the 's are rational functions of which are, from the method of their formation, independent of the particular value of selected. We may therefore write where is a polynomial in and with coefficients rational in . It is evident that for every value of other than . In particular

It follows that the roots of the equation in are . We have therefore where is the coefficient of in , and and are the coefficients of and in . Equating the coefficients of on the two sides of this equation, we obtain where is the coefficient of in . Thus the theorem is proved.

14. We can now apply Lemma (3) of §11; and we arrive at the final conclusion that if is algebraical then it can be expressed in the form where are rational functions of .

The most important case is that in which where is rational. In this case

But Eliminating between these equations, we obtain an equation where is a polynomial. It follows from Lemma (2) of §11 that this equation must be satisfied by all the roots of (1). Thus (4) is still true if we replace by any other root of (1); and as (2) is still true when we effect this substitution, it follows that (3) is also still true. Integrating, we see that the equation is true when is replaced by . We may therefore replace by , being any primitive th root of unity. Making this substitution, and multiplying by , we obtain and on adding the equations of this type we obtain Thus in this case the functions all disappear.

It has been shown by Liouville[37] that the preceding results enable us to obtain in all cases, by a finite number of elementary algebraical operations, a solution of the problem 'to determine whether is algebraical, and to find the integral when it is algebraical'.

15. It would take too long to attempt to trace in detail the steps of the general argument. We shall confine ourselves to a solution of a particular problem which will give a sufficient illustration of the general nature of the arguments which must be employed.

We shall determine under what circumstances the integral is algebraical. This question might of course be answered by actually evaluating the integral in the general case and finding when the integral function reduces to an algebraical function. We are now, however, in a position to answer it without any such integration.

We shall suppose first that is not a perfect square. In this case where and if is algebraical it must be of the form Hence or

We can now show that is a polynomial in . For if , where and are polynomials, then , if not a mere constant, must contain a factor and we can put where and do not contain the factor . Substituting this expression for , and reducing, we obtain Hence must be divisible by . Suppose then that where is prime to . Substituting in the equation last obtained we deduce which is obviously impossible, since neither , , nor is divisible by . Thus must be a constant. Hence where is a polynomial.

Differentiating and clearing of radicals we obtain Suppose that the first term in is . Equating the coefficients of , we find at once that . We may therefore take so that

From (1) it follows that is divisible by . But is not a factor of , as the latter is not a perfect square. Hence either (i) and differ only by a constant factor or (ii) the two quadratics have one and only one factor in common, and is also a factor of . In the latter case we may write where , . It then follows from (1) that Hence is divisible by . Dividing by we obtain and so , which is untrue.

Hence case (ii) is impossible, and so and differ only by a constant factor. It then follows from (1) that is a factor of ; and the result becomes where is a constant. It is easily verified that this equation is actually true when , and that The formula is equivalent to

There remains for consideration the case in which is a perfect square, say . Then must be rational, and so .

As a further example, the reader may verify that if then[38]

16. The theorem of §11 enables us to complete the proof of the two fundamental theorems stated without proof in ii., §5, viz.

(a) is not an algebraical function of ,

(b) is not an algebraical function of .

We shall prove (b) as a special case of a more general theorem, viz. 'no sum of the form in which the coefficients are not all zero, can be an algebraical function of '. To prove this we have only to observe that the sum in question is the integral of a rational function of . If then it is algebraical it must, by the theorem of §11, be rational, and this we have already seen to be impossible (iv., 2).

That is not algebraical now follows at once from the fact that it is the inverse function of .

17. The general theorem of §11 gives the first step in the rigid proof of 'Laplace's principle' stated in iii., §2. On account of the immense importance of this principle we repeat Laplace's words: 'l'intégrale d'une fonction différentielle ne peut contenir d'autres quantités radicaux que celles qui entrent dans cette fonction'. This general principle, combined with arguments similar to those used above (§15) in a particular case, enables us to prove without difficulty that a great many integrals cannot be algebraical, notably the standard elliptic integrals which give rise by inversion to the elliptic functions.

18. We must now consider in a very summary manner the more difficult question of the nature of those integrals of algebraical functions which are expressible in finite terms by means of the elementary transcendental functions. In the first place no integral of any algebraical function can contain any exponential. Of this theorem it is, as we remarked before, easy to become convinced by a little reflection, as doubtless did Laplace, who certainly possessed no rigorous proof. The reader will find little difficulty in coming to the conclusion that exponentials cannot be eliminated from an elementary function by differentiation. But we would strongly recommend him to study the exceedingly beautiful and ingenious proof of this proposition given by Liouville[39]. We have unfortunately no space to insert it here.

It is instructive to consider particular cases of this theorem. Suppose for example that , where is algebraical, were a polynomial in and , say When this expression is differentiated, must disappear from it: otherwise we should have an algebraical relation between and . Expressing the conditions that the coefficient of every power of in the differential coefficient of (1) vanishes identically, we find that the same must be true of (1), so that after all the integral does not really contain . Liouville's proof is in reality a development of this idea.

The integral of an algebraical function, if expressible in terms of elementary functions, can therefore only contain algebraical or logarithmic functions. The next step is to show that the logarithms must be simple logarithms of algebraical functions and can only enter linearly, so that the general integral must be of the type where are constants and algebraical functions. Only when the logarithms occur in this simple form will differentiation eliminate them.

Lastly it can be shown by arguments similar to those of §§11—14 that are rational functions of and . Thus , if an elementary function, is the sum of a rational function of and and of certain constant multiples of logarithms of such functions. We can suppose that no two of are commensurable, or indeed, more generally, that no linear relation with rational coefficients, holds between them. For if such a relation held then we could eliminate from the integral, writing it in the form

It is instructive to verify the truth of this theorem in the special case in which the curve is unicursal. In this case and are rational functions , of a parameter , and the integral, being the integral of a rational function of , is of the form where are rational functions of . But may be expressed, by means of elementary algebraical operations, as a rational function of and . Thus are rational functions of and .

The case of greatest interest is that in which is a rational function of and , where is a polynomial. As we have already seen, can in this case be expressed in the form where and are rational functions of . We shall suppress the rational part and suppose that . In this case the general theorem gives where are rational. If we differentiate this equation we obtain an algebraical identity in which we can change the sign of . Thus we may change the sign of in the integral equation. If we do this and subtract, and write for , we obtain which is the standard form for such an integral. It is evident that we may suppose to be polynomials.

19. (i) By means of this theorem it is possible to prove that a number of important integrals, and notably the integrals are not expressible in terms of elementary functions, and so represent genuinely new transcendents. The formal proof of this was worked out by Liouville[40]; it rests merely on a consideration of the possible forms of the differential coefficients of expressions of the form and the arguments used are purely algebraical and of no great theoretical difficulty. The proof is however too detailed to be inserted here. It is not difficult to find shorter proofs, but these are of a less elementary character, being based on ideas drawn from the theory of functions[41].

The general questions of this nature which arise in connection with integrals of the form or, more generally, are of extreme interest and difficulty. The case which has received most attention is that in which and is of the third or fourth degree, in which case the integral is said to be elliptic. An integral of this kind is called pseudo-elliptic if it is expressible in terms of algebraical and logarithmic functions. Two examples were given above (§ 10). General methods have been given for the construction of such integrals, and it has been shown that certain interesting forms are pseudo-elliptic. In Goursat's Cours d'analyse[42], for instance, it is shown that if is a rational function such that then is pseudo-elliptic. But no method has been devised as yet by which we can always determine in a finite number of steps whether a given elliptic integral is pseudo-elliptic, and integrate it if it is, and there is reason to suppose that no such method can be given. And up to the present it has not, so far as we know, been proved rigorously and explicitly that (e.g.) the function is not a root of an elementary transcendental equation; all that has been shown is that it is not explicitly expressible in terms of elementary transcendents. The processes of reasoning employed here, and in the memoirs to which we have referred, do not therefore suffice to prove that the inverse function is not an elementary function of . Such a proof must rest on the known properties of the function , and would lie altogether outside the province of this tract.

The reader who desires to pursue the subject further will find references to the original authorities in Appendix I.

(ii) One particular class of integrals which is of especial interest is that of the binomial integrals where , , are rational. Putting , and neglecting a constant factor, we obtain an integral of the form where and are rational. If is an integer, and a fraction , this integral can be evaluated at once by putting , a substitution which rationalises the integrand. If is an integer, and , we put . If is an integer, and , we put .

It follows from Tschebyschef's researches (to which references are given in Appendix I) that these three cases are the only ones in which the integral can be evaluated in finite form.

20. In §§7—9 we considered in some detail the integrals connected with curves whose deficiency is zero. We shall now consider in a more summary way the case next in simplicity, that in which the deficiency is unity, so that the number of double points is It has been shown by Clebsch[43] that in this case the coordinates of the points of the curve can be expressed as rational functions of a parameter and of the square root of a polynomial in of the third or fourth degree.

The fact is that the curves are the simplest curves of deficiency . The first is the typical cubic without a double point. The second is a quartic with two double points, in this case coinciding in a 'tacnode' at infinity, as we see by making the equation homogeneous with , writing for , and then comparing the resulting equation with the form treated by Salmon on p. 215 of his Higher plane curves. The reader who is familiar with the theory of algebraical plane curves will remember that the deficiency of a curve is unaltered by any birational transformation of coordinates, and that any curve can be birationally transformed into any other curve of the same deficiency, so that any curve of deficiency can be birationally transformed into the cubic whose equation is written above.

The argument by which this general theorem is proved is very much like that by which we proved the corresponding theorem for unicursal curves. The simplest case is that of the general cubic curve. We take a point on the curve as origin, so that the equation of the curve is of the form Let us consider the intersections of this curve with the secant . Eliminating , and solving the resulting quadratic in , we see that the only irrationality which enters into the expression of is where

A more elegant method has been given by Clebsch[44]. If we write the cubic in the form where , , , are linear functions of and , so that , , are the asymptotes, then the hyperbolas will meet the cubic in four fixed points at infinity, and therefore in two points only which depend on . For these points Eliminating from these equations, we obtain an equation of the form where , , are quadratics in . Hence where is a polynomial in of degree not higher than the fourth.

Thus if the curve is so that being an imaginary cube root of unity, then we find that the line meets the curve in the points given by where and

In particular, for the curve we have

21. It will be plain from what precedes that can always be reduced to an elliptic integral, the deficiency of the cubic being unity.

In general integrals associated with curves whose deficiency is greater than unity cannot be so reduced. But associated with every curve of, let us say, deficiency there will be an infinity of integrals reducible to elliptic integrals or even to elementary functions; and there are curves of deficiency for which all such integrals are reducible.

For example, the integral may be split up into the sum of the integral of a rational function and two integrals of the types and each of these integrals becomes elliptic on putting . But the deficiency of is . Another example is given by the integral[45]

22. It would be beside our present purpose to enter into any details as to the general theory of elliptic integrals, still less of the integrals (usually called Abelian) associated with curves of deficiency greater than unity. We have seen that if the deficiency is unity then the integral can be transformed into the form where[46] It can be shown that, by a transformation of the type this integral can be transformed into an integral where

We can then, as when is of the second degree (§3), decompose this integral into two integrals of the forms Of these integrals the first is elementary, and the second can be decomposed[47] into the sum of an algebraical term, of certain multiples of the integrals and of a number of integrals of the type These integrals cannot in general be reduced to elementary functions, and are therefore new transcendents.

We will only add, before leaving this part of our subject, that the algebraical part of these integrals can be found by means of the elementary algebraical operations, as was the case with the rational part of the integral of a rational function, and with the algebraical part of the simple integrals considered in §§14—15.

FOOTNOTES: