596 Mr. AIRY'S supplement TO A PAPER ON THE INTENSITY OF LIGHT 



" University College, London, March 1], 1848. 



" In reply to your request that I would send you a sketch of the method which I 



communicated to you some years ago, for finding the numerical value of J^ cos (w' - mw) dw, I 



send you the following. I am not aware that there is anything about it peculiarly my own, or 

 other than what would suggest itself as a matter of course to any one familiar with the current 

 methods in definite integrals. 



" The series which I furnished depend ultimately upon the following formula : — 



r<t. „ „ , . „ COS JlO 



r e-rcoss.u- COS (r Sin 0.W). «<;""' rfMJ,= r„ , 



r^ _ sin nQ 



I g-rcose.K. gi„ (,.sin 9.(4,) . u!'-' dw = r„ , 



Jo ^ ' y 



in which r and 9 are independent of w, r cos d is positive, n is positive, and F,, stands for 



J' e~''x"~^dx, as usual. Under these conditions the theorems do not or need not rely upon any 



notion of algebraical as distinguished from numerical equality. Calling either of them f^w .dw, 

 common arithmetical calculation would establish any degree of approximation between the conver- 

 gent series 00 . a + (pa. a + d)2a . a + ... and the asserted value of the definite integral, if a were 



TT 



taken small enough. And this for any value of 0, from ^ = to = --/3, /3 being of any 



degree of smallness. But when = - , the nunKrical character of the equivalence is lost, and the 



equations assume the same character as 1 - 1 + 1 - 1 + = \, and are subject to the same 



discussion. 



"The above equations were first obtained by substituting a +b\/- 1 for a in 



•I" a" 



which is an equivalence of numerical character even after the substitution, if a be positive, and h 

 (be it positive or negative) numerically less than a. For the use of the expansions ot e-'"'\/^l and 

 (a + 6 y/ITi)-" in powers of b would produce an equivalence such as 



Ao + A,k + A^le' + = B, + B,k + B,k' + 



where k = %/- 1, A„ = B„ is a numerical equivalence, and 2J„ is a convergent series. But, 

 when 6 is numerically greater than a, a convergent series would be rendered divergent in inte- 

 gration: and, when this happens, I do not see any way to place the divergent series so obtained 

 upon the same footing as those of ordinary algebra. 



" It is not however necessary to depend upon this introduction of divergency. If we call the 

 two integrals C„ and S„, and differentiate both with respect to Q, we have 



'- = !• sin 9 . C I - r cos Q . >S'„+ 1 , 



— ^ = r sin . A'„+, + r cos . C„t,; 



