114 



PROFESSOR G. H. DARWIN ON THE INTEGRALS OF THE 



Accordingly if there is a term A n [/r" + 2 cos 2 " + * ( ' in j K~ cos 2 B ($,*) 2 ^ and a term 



fc' J *sin 2 *^ -^ in ((C') 2 ^> then there must be a term 



cos 



8 " 



in 



[(JjJ/) 8 , and a term ( }" A ,,U' 2 " + - sin 2 " "^> -2. It follows that the coefficient of 



8/r 5 cos B cosy 

 s)-f. s . ni f Ms 



|K-"' ' = COS'" - - ' |K"-'"' Hill*" ,/> '^ - (-) + ' f * 5 " COS 5 " ^ f ' 2 " + 2 sin 2 " + * <f> <** . 



For the snko of hrcvity I call this function [2?t + % 2, 2m], and we may state that one 

 term in the required expression is ( )'"A a A, a [2 + 2, 2m], where [ ] indicates the 



T , i x 8& ?1 COS B COSy 



above [unction 01 the tour inteffrals. It follows that 1* (cos) -=- .- ., n 



sin' B 



A- 



.!,,/ 4,0]- A? 4, 2]-f .1,J.,[4, 4 



. i 



0. o ,-1,/J, [<;, 2 + A." [r,, 4 ] . . . 



.... (1). 



Since 



[2n, 2m] = [*-" co8~"0 (W jV-"' sin 2 '" f /> f/ f - (-)"-'" f^'" sin 2 '" 6> fW f/c' 2 " sin 2 "</> ^, 



it is clear that 



[2>,, 2m] = -(-)"-'" [2,n, 2ft]. 



Hence if n and ??i differ by an odd number [2n, 2m] = [2m, 2ft], and if they differ 

 by an even number [2n, 2m] = [2 in, 2n\. Also \2n, 2n~] = 0. 



c 1 ." lift r iir r!A\ 



T J_ " 1 f n ^ i On /I tvv i' r\ ~) / /-^ji " OK I tt'li) i | 



Let us write (2ij = /c" cos 2 " ^ . !2; = K -" sin 2 " </> --L, so th; 



.'o A Jo r 



[2n, 2m] = {2n} {2m}' ( )"-'" {2n}' {2m}. 



We must now evaluate these functions. 



Since A 2 = 1 K- sin 2 0, we have bv differentiation 



at 



{ (-2n - 1 ) K 2 cos-" - (-2n - 2) (/c 2 - *' 2 ) cos 2 2 - (2ft - 3) *' 2 cos 2 " -* 0}. 

 Integrating between \-n and and multiplying by /c 2 "" 2 we have 





