346 Mil MURPHY'S THIRD MEMOIR ON THE 



Put now for m successively every integer from 1 to w inclusive, and 

 take the sum of all the definite integrals thus resulting, hence 



/ecos»0h.l. {(l-2Acos0 + A^)(l-2Acos2e + A^)...(l-2Acos«0 + A*)} 



\n , w' -^, w" -4, 1,1 



= - ttX- .h + - .h" + — . A" + ...- . h"). 

 \n n n » J 



Now the quantities -, — , — , &c. are the reciprocals of all the 



Tt ft Ti' 



possible divisors of n, and therefore this definite integral may also be 

 expressed by 



"^ -^{k + -,h''' + \h"" + ...-h"}. 



'^ n' n n ' 



For 9 in the preceding equation write 20, the limits of the latter 



variable will be and - . 



2 



Also put h = 1, and therefore, 



1 -2hcosd + h' = 2{l-cos2(p) = 4!sm^(f), 

 1 - 2A cos 20 + ^2 = 2 (1 - cos 40) = 4 sin' 20, 

 &c. ; 

 .-. h.l. {{l-2heos9 + h') {I -2hcos2e + h'')...{l - 2hcosne + h')} 

 = 2w h. 1, (2) + 2 h. 1. {sin sin 20. ..sin w0}. 



The integral of the constant multiplied by cos2«0 vanishes, and therefore 

 7^ h.l. {sin sin 20 sin 30. .. sin »0} , cos2»0 = — t|~ + — +— , +...4-ll; 

 and multiplying both sides by , we get this theorem. 



The sum of all the divisors of a given number n, including the 

 number itself and unity, is expressed by the definite integral 



4t7l 



/^h. 1. {sin sin 20 sin 30. .. sin w0} . cos2w0. 



