THE ORBIT OF URANUS. 15 



in the general term of the above scries for sin g its value in imaginary 



exponential functions 



2 sin = i/ IT (c~' s/ - ri c*^~-i) 



we find by the binomial theorem, using the notation of combinations, 

 ', n( I) (n + !) n! 



1.2.3. .. 



_ 1)- tfc< U*^ _|_ (_ 



n+l 



*=i \ 

 j 



Differentiating n 1 times with respect to g, and putting together the first and 

 la>t terms, the one after the first, and that before the last, and so on, we find 



C(n I)"- 1 (c< "' "= -[- <r<"- 1 )' ^) + etc. 

 +i 



Sul)stituting for the exponentials their values in circular functions, and dividing 

 1>\ % J" +1 we have 



9*-' sin" +1 fl 1 f > 



(u + 1)-' cos (n + \)g-C(n - I)- 1 cos (n - \)g 



< ij A \ +! 



+ C(n 3)' cos (n 3) or etc. 1 

 *f ) 



the series terminating at the last positive coefficient of g. Substituting this last 

 value in the general term of the scries which gives cos , we have 



COM 



u=T * {(n+l)"- 1 cos(n+l)( 7 -C'(M-l)- 1 cos(-l) fl r+etc. 1 

 n-o n I f, ( 4 1 J 



Let us now substitute for n another variable i, putting in the first term of the 

 tactor t = n -(- 1, in the second i = n 1, in the third i = n 3, etc. The 

 limits of finite integration with respect to i will then be 



in the first term, -f-1. to -j-oc, 



in the second term, 1 to -\-cc, 



in the third term, 3 to -|-oc, 



etc. etc. 



But all the coefficients of g will then be t, and the formula supposes the factor of 

 cos ig to vanish whenever t is zero or negative; whence, those elements of the 

 finite integral in which t is negative must be omitted, and all the terms must be 

 taken between the limits -|- 1 and -f- oc. Making the proposed substitution we 

 have 



