116 PROFESSOR G. H. DARWIN ON THE INTEGRALS OF THE 



When these functions are introduced into (1) and the terms re-arranged, I find : 



T i v 7J- cos /3 cos y 

 If (cos) 4- - - T-B~ ~ = 

 sm 3 /3 



A* - M*A* + i 

 + 2 (,c 2 - /c' 2 ) [-A. ^ _ ^ K WA,A Z + - 6 -? 7 . KVM 



+ linr CH 2 - 4 ) - 3W;M 4 3 - srb ti (4-6) - 



+ 57779 [* ( 6 - 8 ) - 7W *] *V*,M, - - 7 - e n -[- (H. 10) - 9 2 *V 2 J VM A + . . . 



' + 525*'^) A'J.i. - (2). 



Li this result a good'many terms are added which are not deducible from the table 

 of functions given above, but every term as stated here has actually been computed. 

 The laws governing the succession of terms in the first six lines seem clear, but I do 

 not claim that the proof of the laws is rigorous. I do not perceive how each series 

 is derived from those preceding it, and I have no idea how the series beginning with 

 J,,,!.!. would go on. With sufficient patience it would no doubt be possible to 

 determine the general law of the series, but I do not propose to make the attempt at 

 present, since we have more than enough for the immediate object in view. 



This result (2) is, of course, equally applicable to the integrals of the type //* (sin). 



In order to effect the required integrations we must define the functions, and I 

 take the definitions (with a few very slight changes) from '2 of " the Pear-shaped 

 Figure." In order to use the preceding analysis it is necessary that the square of the 

 P- function and the square of the cosine or sine function should be the same functions 

 of K- cos 2 9 and of K'~ sin 2 (j>. But as in the definitions to be used this symmetry 

 does not hold good, a difficulty arises, which may, however, be easily overcome. If 

 the P-function be multiplied by any factor /, and the cosine or sine function by any 

 factor g, the integral will be multiplied by f' 2 g~. I therefore introduce such factors 

 j and (j as will render the residual factors of the squares of the P and cosine or sine- 

 functions symmetrical in the proper manner. 



It seems desirable to show how the results found here accord with the approximate 

 integrals as found on pp. 548-9, 22, of " Harmonics." In this connection I 



remark that ~~'9a~~ f when written in the notation of "Harmonics," is 



V (z/ 3 I) 1 ( v 1 - LLE) a factor which I denoted in that paper by M. 

 \ 1 pi 



It does not seem necessary to give full details of the analysis in the several cases, 



