128 



PROFESSOR G. H. DARWIN ON THE INTEGRALS OF THE 



If these expressions be developed in powers of */ I find, on reintroducing the 

 factor K f ~f 2 y 2 , 



- (1 - '' K'~ + 241 'i\ K '-2 



3 * ' ^ 4 2 5 6 



J 1 



= y M. 



- 



7 sin 3 /3 3. 5* ' 

 - S/8 + 



os p cos ; 

 7 sin 3 y8 9 



, ,1 , A ,, , 8 , /4 , 



.5 K V 2 * T^ 256 K / K J (I ) 



In " Harmonics" I defined 



But we have defined it above by 



i/ 



P, 1 (p.) = / ( ] - ,c- sin- 5)5 (^ sin- 5 - ,/). 



Therefore 



. J + 



+ a ^ + .^ ^ 



^ 



Whence / = '/ ( I + y /3 + ^ fr). 



This value also satisfies the expression for //<-. 

 Again I defined 



= COS <, - - 



= cos 1 



(!OS 



cos 



. . (49). 



But we have defined it above by 



(]' = (jK r cos (f> (K'~ cos 2 (f) ry'' : ). 

 Therefore 



With the above value for (/ we have </- = f (I + J-^ -f -fa fP) ; whence 



