THE ORBIT OF URANUS. 15 



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

 exponential functions 



2 sin (/ = /^n" (c-^^^i — c^>^^^) 



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



*,^ n (n — 1) (n — ,9 + 1) _ n ! 



»~ 1.2.3... 5 ~sl(n — s)] 



f 3 o 



2" + 'sin" + '<7 = (y l)""*"' \ c~<" + ^'«' ' -'f ^c-(«-i)ff''-i I ^g~(n-3)<7i'^rT 



Differentiating /i — 1 times Avith respect to g, and putting together the first and 

 last terms, the one after the first, and that before the last, and so on, Ave find 



— C(n — ly-' (c("-')^''^ -1- c-f"-i)^ ''^) + etc. 



Substituting for the exponentials their values in circular functions, and dividing 

 by 2"+^ Ave have 



^^i~^ - - -gv { ('* + 1)"" cos (n + 1):/ - C^(u - 1)"- cos (n - 1) J/ 



I C(7i — 3)"-' cos (m — 3) <7 — etc. 1 



n+l i 



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

 value in the general term of the series Avhich gives cos u, we have 



cos u ="2^ ^" I (n -f 1)"-^ cos (n + 1) <7 — C (w — 1)"-^ cos (n — l)g-\- etc. | 



n=0 n ! 2" ( n+l J 



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

 last factor i ^ n -\-\, in the second i = ?i — 1, in the third i^^n — 3, etc. The 

 limits of finite integration Avith respect to i Avill then be 



in the first term, -|-1 to -\-cc, 



in the second term, — 1 to -[-oc, 



in the third term, — 3 to -|-oc, 



etc. etc. 



But all the coefficients of g Avill then be i, and the formula supposes the factor of 

 cos i g to vanish Avhenever { is zero or negative; Avhence, those elements of the 

 finite integral in Avhich i is noijative must be omitted, and all the terms must be 

 taken betAVcen the limits -f- 1 and -\- ex. Making the proposed substitution we 

 have 



