Mr. Ivory's Reply to Professor Airy. 19 



resulting from Laplace's development, and that obtained by 

 the algebraic process. Each of these expressions is obtained 

 in a manner that is unique: there is nothing uncertain in 

 either ; and it must be admitted that they are identical. Taken 

 in toto, they both consist of the same simple quantities con- 

 nected with the same signs ; but in one, these quantities are 

 distributed in groups possessed of a general property ; and in 

 the other, there is no artificial arrangement. If one be nu- 

 merically equal to the finite quantity y, the other must be so 

 too; and if the first may be substituted for j/ in any investi- 

 gation, so likewise may the second. 



It is evident that the radicals expanded in the algebraic 

 value of J/ produce, converging serieses only; and that, by 

 extending the portions of the serieses taken in, we may ap- 

 proximate to the value of j/ indefinitely. When fi, = + i, 

 tlie approximation is not di sturbed by that part arising from 

 the expansion of a/ 1 — /«.' and its powers ; and the jjart pro- 

 duced by — =rr^ and its powers, which*might be infinitely 



great, vanishes, because every term is multiplied by s, t which 

 are equal to zero when ju. = + 1. Thus the approximation 

 holds good for all values of ,a between the limits +1. I there- 

 fore conclude generally that every function of two arcs, which 

 is always finite between the prescribed limits, may be changed 

 into a finite and rational function of three coordinates of a 

 sphere, that shall approximate in any required degree to the 

 given function. 



There is no difficulty attending this analysis when y is ac- 

 curately a finite and rational function of three coordinates of 

 a sphere ; and we may certainly comprehend in the same con- 

 clusion all cases in which we can approximate indefinitely to 

 the value of y by expressions of the same kind. We thus 

 obtain the theory in all its generality, and we place it on its 

 right basis, which is the nature of the development. 



I have never found fault with the demonstrations of the 

 theorem, except on just grounds. In the Mccaniquc Celeste, 

 livr. ii. No. 10, the thickness of the molecule at the attracted 

 point is not considered; which occasions a difficulty of which 

 Lagrange has treated : and in livr. xii. the same thickness is 

 made ultimately divisible by the s([uare of its distance from 

 the attracted point; which would limit the ajialysis to a very 

 particular class of functions. Tlie demonstration of M. Pois- 

 son amounts only to this, 



which seems an identical proposition, the quantity into which 



n 2 y is 



