USE OF DEMOIVEB'S FUNCTION. 47 



2*^' = x{ V- y-') - y ( F'-' - V) + J ( V'- V') - + 

 = i(i + xV)- z(i + .ry"') 

 = I ^ "^^^ 



1 + xV-' 

 Now, denoting sin 6 hj s and cos (9 by c, 



9 



j6'= Z(l + o; + ("s.t) — ^1 + ex — isx) 



i l + cxi I 1 + cxi 



= 2i\ ^'^ - I. 

 ll + cx ^ 



"t" x- v^. — : r-, - + 



+ cx (1 + cxy " (1 + CX) 



X sill d 



'?)■ +nr,-l 



= 2i tan 1 , ^ 



1 + X cos 6* 



Or, thus — 



]fZi = /]i + ^^^' ^ = ¥28 = cos 2,S' + i Bin 2,S. 

 l + a:F-' 1 + 2a; cos 6» + a:^ 



• ■ • ^^^ = — ,. , „ T. TT = coa S + i sin &', 



v/{14-2a;coséy+ ar^ 



Hence equating real, and also imaginary parts, 



a: sin 



Sin S = 

 CoaS — 



^\l + 2xcoad + xr} 



1 + a; cos 

 x/^l-F2a;cos(9+ar^ 



1T7-1 . r, X sin ^ , „ , a; sin 6* 



Whence tan k, =: -^ , ^ ^ » and A5=tan-' 



1 + a; cos (^ '*"" "—"''" 1 + a; cos y 

 OC sîii ^ 



Ex. 6. To expand :j „ , ^ _i_ ^4 in ascending powers of x, and multiples of (9. 



1 — •Ji.L COS f/ ~|~ «C 



ihis expression is • 



2? (1 - x^'V) (1- jrF-i) 



Assuming --^L^^tZJ!) = -^^ + ?^_, 



° (1 - :/.-=F) (1 - jfV-') — 1 _ ar F 1 - ar'F"' 



we readily find, after the manner of partial fractions, that A—V, and 5= -V'\ 



.-. (i_y^^~^"^^^-.-, = xiV-r-^) + a;^(P-F-) + a^ ( F^ _ F-) • • • 



= 2i Ix sin (9 + .r- sin 2« + a;' sin 3^ + . .} 

 and, dividing by 2i, 



X a'ln t) o „,, - . „ 



1 - 2x^ cos^ + X- = •''■ «'^ ^ + ^ '''^ 2^ + -^'^ sin 3éy + ■ • • 



Ex. *7. Griven tan q> ^ n tan ^ to express cp in terms of the functions of tj and its 

 multiples. 



„ Fy - F~V _ nVH - nV^'fi 



