134- On a new Theorem in the Calculus of Finite Differences. 

 cos -rcfl CQS n . _ »cos"- a r n. n - 3 cos"" 4 



»«— 1 -. -o "T 



n-6 ' ' ^ ' 



w . ra — 4 . ra — 5 cos" 9 



12 3 ' 8 s 



The expansion of cos n 0, in powers of cos 0, fails when n is 



either negative or fractional, as x n +x~ * cannot be expanded 

 in those cases in powers of z. It is also clear that the expan- 

 sion can contain no negative powers of cos 0, as negative 



powers of z cannot appear in the development of x n -\-x~ n . 



To expand sin n in powers of sin 0. 



The investigation of this development must be divided into 

 two cases, as n is odd or even. First let n be odd. 



Put = — — <J>, then n = — — n <p, cos n = sin n <£>, 



and cos — sin <$. 

 Making these substitutions in (X.), we find 



sinwA . « n sin M ~ <J> n ,n — 3sin w- d> a /VT . 

 -^ nr = sin «> - T — ^ + _ ^— . . &c. (XL) 



Again, when n is even 



Differentiating (X.), and dividing by n d 0, we find 



> . (XII.) 



sinrc0 . . [cos M *0 n — 2 cos w ~ 

 sin 



>« L 2 1 1 



w — 3 . n — 4 cos 



12 2 5 * 



Now let 9 = — — <p, then sin n = + sin w <j>, sin 9 = cos <f>, 



cos 9 = sin<f>. Making these substitutions in (XII.), there 

 results 



_ sin n<p fs'm n ~ l <p n — 2 sin w_3 d> 



+ = cos d> < - ~ 1 



2 W L 2 1 1 2 3 



• n ~ 3 ' n ~" 4 sin ra ~ 5 4 > 

 + " 12 2*~~ ' 



the upper sign being taken when n is divisible by 4, if other- 

 wise, the lower. 



