] 8 Mr. Ivory's Itepty to Professor Airy. 



carious and contained between given limits. I conclude, there- 

 fore, that on the grounds we now go upon, we have, 



y = Y/^'i + Y''^ + Yi^\ + &c. (A) 



with all the generality that Laplace and Poisson have asserted. 

 Supposing that « is less than unit, let us expand the inte- 

 gral expression, thus ^, 



1 riy^)!i± ^ Yn + ccxn + u\Yr-) + &c. 



Here there is numerical equality between the finite expression 

 on one side and the infinite series on the other : for, a being 

 less than unit, the terms of the series continually decrease 

 and finally become insensible. But, when « = 1, the princi- 

 ple by which the equality was before proved disappears, and 

 we can no longer affirm that there is an equation. If we sup- 

 pose a to become ever so little greater than unit, there is a 

 disruption of the continuity, the quantity on one side becoming 

 negative, and the series, on the other side infinitely great. In 

 many cases it is certain that the series, in the circumstances 

 mentioned, is absurd and insignificant in respect of numerical 

 value. How are we to separate such cases from those in which 

 the analysis may be employed as a legitimate means of inves- 

 tigation y It may be argued that the theorem alone is not suf- 

 ficient; because, in the demonstration the quantity j/ is con- 

 sidered as finite; and some check in respect of numerical 

 quantity is always requisite when a finite is changed into an 

 infinite expression. How comes it that a series, which is in- 

 terminable, and in which no principle of convergency has been 

 pointed out, nevertheless represents a finite quantity with nu- 

 merical exactness ? 



Suppose that j/ is a rational function of cos 6, sin Q, cos w, 



sin w, or of f*, v^ 1 — (x", cos w, sin w ; and put, 



f = V 1 — !"-'• cos CO, 5 = -v/ 1 — jw-^. 



then cos <o = — -n-=-, sin w = 



By substituting these values it is evident that j/ will be con- 

 verted into a function of fj-, s,t; and, by expandhig the radi- 

 cals wherever they occur, it will be changed into an infinite 

 expression which is a rational function of j«., .*:, /. This ex- 

 pression is unique ; for the algebraic operations can be per- 

 tbrmed only one way. We have now two infinite expressions 

 of 1/ in terms of the same quantities fx., s, f; namely, that (A) 



* Conn, des Terns 1829, p. 333. 



resulting 



