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ROYAL SOCIETY OF CANADA 



where c stands for cos ^and .s for sin 6. Then equatinii- real and iinao-in- 

 ary parts, 



C=e^^''^'^^^cof^(6-^s\n-6); 



S = e 



sin f> cos a 



sin (^ + sin ^6). 



4. To find the genei-ating functions of the two complementary series, 



" = ^^ cos nd cos '^6 '^ ^, ^ sin n^ cos ""O 

 2, , and -^ ; . 



n=l 



n = 1 



n ! 



cV 

 = e _\ 



we have, adopting the preceding notation, 



= / .e'''-l = /- Fc-s - 1. 

 And equating real and imaginary parts, 



C = e ^<^^ "^ cos (cos ^ sin ^) — 1, 

 aS' = e. ^o« ''^ sin (cos ^ sin 6). 



5. In developing cos nB in powers of the cosine of 6 we have 

 primarily the development, 



cos nS = r" - "rv^'^-^s- + ''C.^^'^s^ -+..., 



in which c stands for cos 8 and s for sin i'^. 



As the sines are all in even powers, the right-hand expression may 

 be written rationally in terms of the powers of cos ^. The development 

 as thus effected is, however, not only laborious but also impracticable. 



Second!}', we maj'put x -j-—^=2 cos 0, and expand the expression 



( 1 + ~ .'• 1 ( 1 + -^ ) = 1 4- 2c cos fi + Z-. By taking logarithms of 



both sides of this identity, and ])ieking out from the expansion of these 

 logaritlims the coettieient of 5" and equating, we obtain 



cos nO = 2 cos H — n . 2 



-+ • . . 



The picking out of the parts of the complex coefficient of ~" from the 

 right-hand member is the laborious ])art of this process. 



Thirdly let a" + p„ a"~- + ^n«"~'' + r„a"-^ + . . . = 6" 



^,-2 _|_ p^^_^a"-* -f- fin-M'"^' + • ■ • = ^>"~^ 



^n-i _j_ p^^_^a"--^ -f . . . = 6"-' 



a"-^' -\- . . . = /y'-« 



n—5 n— 2„ , n (n — 3) «-5 n— 4. 



cos & -\- ^-j 2 cos a 



