48 N. F. DUPUIS ON DEMOIVEE'S FUNCTION. 



whence by reduction we easily obtain, 



_ (1 + n) V2H + (1 - ») 

 •'-^^(1 _|.„) + (l-n) .¥20 



1 + VI V~'iO 1 - n 



= ^-^- l + mF2éy"' where m = ^^^ 



Taking logarithms — 



•2i cp—2i6- 2i m sin 26 + 2i. ''"l gi^ 4,^ _ + . . 



. • . (p=^6 — m sin 2fi -\- -y sin id h • • • 



Ex 8. To develope cos nO cos"H and sin nH cos"^ in powers of tan 0. 



Take cos"^ VîiH 



Then co."^ Fn^ = [y^,J = (^_^-^^-— ^J 



= (!-«■ tan 6)"" 



n(w+ 1) „ „ n(M+ 1) (n + 2) , 



= 1 + nj tan f^ + ., , ^ i= tan-^^ + 3, — ^ r tau-f^ + . . . 



and equating real parts, and also imaginary parts, gives — 



Cos-^ cos ne = l — ".a; tan=^ + -^^ tan^i^ - + . . . 

 And Cos"^ sin nB = n tan - "^3 tan^l^ + "l?; tan=W h • • 



where "H, denotes the number of homogeneous terms of r dimensions which can be 

 made from n letters and their powers. 



Ex. 9. G-iven sin ^ = -r- sin {A-VC) to develope the angle A in terms of the func- 

 tions of C and its multiples. 



Here, VA -V''A = ^V{A+G) - ^V'\A+C). 



^^VA-ro- ^r'"A-r~'c. 



Whence F2 4 = 



And taking logarithms — 





a . _. _ . a 



2iA = 2i y sin C + 2i 5^, sin 2G + . . 



a ^ a- ^ «' „ 



vl = y sin (; + 9T2 sin 2(7 + 0T3 sin 3(7 +. . 



