106 Mr, Adams on [Feb. 



■ri i. COS. * \ — a x^ + c X* — e x^ + ffx* -^ &c, 1 



For cot. X = = = - 



sin. X X — b x^ + d x'> — fx^ + h X* — &c. x 



-^ (Aa? 4"B x' + C x^ + D x7 + E j;» + &c.), radius unity. 

 From whence we obtain 



^ax^ + cx^-ex^+gx^-kc,^^ (-C + IU-A rf-/):i« 



&c 



u 



Then by equating the homologous terms, we have 



B = + A i - (c- - d) = -1, 



C = - A rf + B 6 + (e -/) = 4 



J)= +Af-Bd+Cb-(g-h) = jl-., 



&c 



Where the law of continuation is evident, we therefore, have 



COt.X=--(j + - + — + _ f &c.) ^^ 



In like manner we may develope the sec. x. ^ j.^, f^^„■^l 



sec. X — = •; ; 1 — J— = 1 + Ax- + 



COS. a- \ — a X* -\- c x^ — ex^ + g X* — &c, 



B a:* 4- C jc^ 4- D a:» -F &c. radius unity ; from which equation 

 we obtain the following, viz. 



n 



'(A — a) x"^ 

 (B — A fl + c) x^ 

 = ^ (C - B a + A . f - e) a;« 



I (D - C a + B c - A e + g) 



L&c 



Then by making the coefficients of a;-, x^, x^, x*, &c. respect- 

 ively equal to nothing, we shall have 



A = a = -, 



2 



B = A a - c = 24", 



111 

 C = -Ac + Ba-e = — , 



'vtK ^Oit'srH Hi! ^ 8064' 

 &c. 



The law of continuation being evident, therefore, 

 ■ec.x = ] + ^-4- -^ + -^ + ^5^ + &c. 



