486 PROFESSOR MOSELEY, ON THE THEORY OF 



Now, 



(l+a)cos9 — cos^ — sin^taiTP a cos — (1 — cos9) + l -cos^ —sin 4' tan* 

 tan 4' ~ tan "I' 



9/ ^ ^\ 4' 



= a cos 9 cot S' — sin' — cot ——tan — - tan — 



2V 2 2/ 2 



= a cos 9 cot 'i' — sin* — cot — — cos^ — tan — . 

 2 2 2 2 



Substituting this value for the first member of the preceding equa- 

 tion ; multiplying by the denominator of the fraction in the second 

 member, and neglecting powers of a above the first, 



— cos 9 cot ^P - -jsm- 2 cot - + cos - tan -| ^^ 



= a(cos 'ir - COS 9) — . 



9 



-a{sm g 



cot- + cos=-tan-J{*-9} 



r 



Whence, by transposition and reduction, 



. i,9 ,i' ,9^ * X 



sin 7; cot — + cos' 7; tan - = - 



2 2 2 2/- 



{/■ 9 '4' 9 "i'\ X \ar' 



(cos^-cos 9) + ( sin^-cot - +cos*-tan - j (*- 9) 5-cos9 cots' >^.... (60). 



9 S' 9 '4' 



Assume cos'— tan — +sin' — cot „ = y ; 



,9, i' . , 9 



.-. cos^ 7; tan — — sin' - cot — = •v/w" - sin' 9 ; 

 2 2 2 2*^ 



••• tan % = \\y^y/y'- sin' 9 } sec' % (61)-, 



d tan - ^ {y + y/'it — sin' 9 } sec' — tan - 



-^ ^/-sin'9 cos'® tan I- sin' ^ cot :i: 



2 2 2 2 



