CORRECTIONS.
Reading edition
Page 28, line ii. Prop. III. after e, insert the base being unity.
Page 30, line 8, Cor. 10. for n t u, read n t t; line 9, for product &c., read square of the cosine of incidence.
Page 31, line 5, Cor. 11. for , read .
Page 31. Prop. V. Cor. See the note in p. 60.
Page 33. Prop. VIII. By a mistake of a sign, the eighth proposition is rendered erroneous; no use having been made of that proposition, it has been inserted without proper revision. It ought to stand thus, with its demonstration:
Proposition VIII. Problem.
To find the path of a ray of light falling obliquely on a sphere, of a refractive density varying as any power of the distance from the centre.
The refractive density, in the sense of these propositions, varies as the ratio of the sines, and as the velocity of light in the medium. (Schol. 2. Prop. I.) Let the velocity at the distance be ; then, considering the refractive force as a species of attraction, we have, in Prop. 41. l. 1. Princip. , , the sine of incidence, the radius being unity, , ·, and the fluxion of the area described by the radius ·. Let the sine of the inclination to the radius at each point be called y; then , , and the fluxion of the area = , of which the fluent is , y being the sine of the arc Y; and the angle corresponding is . The value of that angle being found for any two values of x or y, the difference is the intervening angle described by the radius. This angle is therefore always to the difference of the inclinations as to , and the deviation is to that difference as to .
Corollary. Hence, in the passage to the apsis, and the return to the surface, the deviation is always proportionate to the arc cut off by the incident ray produced: therefore such a sphere could never collect parallel rays to any focus, the lateral density being too small towards the surface.
Page 33, line 20, for but the two last &c. read the seventh may either be deduced from the eighth, or may be demonstrated independently of it.
Page 42, line 18, after internally, insert Or, if a lens of equal mean dimensions, and equal focal length, with the crystalline, be supposed to consist of two segments of the external portion of such a sphere, the refractive density at the centre of this lens must be as 18 to 17.
Page 47, line 12, for calculated &c. read estimated by means of the eighth proposition; and probably.
Page 53, line 24, for 24, read 21; line 25, for 17, read 15.
Page 61, line 21, for sixtieth, read fortieth..