488 PROFESSOR MOSELEY, ON THE THEORY OF 



.-. tan|=tan|?-{2sin^|» 



tan*^"-tan^^ 

 2 2 



unt.)|-|}...(63). 



By equation (48), omitting powers of a above the first, 

 P= [X+arH^-e)} cots' (64); 



dP 



.-. j57 = a/^cot*— {X+ar^(4'-e)} cosec*'*', 



also by equation (61), * = 2 tan"' |i (y + \/i/^-sm' 9) sed' -\ ; 

 d^ {l+y(y^-sin^e)-3}sec^| 



di/ _ Q 



l+^\2f + lyVf - sin' e - sin' 9} sec* g 



y + y/y' — sin' 9 



\/f - sin' 9|cos 9 + ly (y + y/y^ - sin' 9) sec' |l 



2 tan -cos' - 



jsin 2 cot - - cos'- tan -[ |cos 9 + sin' - + cos' ^ tan^- [ 



— 2 tan -cos - 



2 2 — sin •^^ 



{9'!' 9'i'lf9 9S'l 9H' 9 



sin 2 ^"* 2 -^os' 2 t3" 2) (cos'' g"*"^"^' 2*^"' 2) ^^"' 2 ^°* 2 ~^°*' 2 *^" 



