Vol. lxxxvi.] philosophical transactions. 709 



and COS. 3s = 1 — 3as; + 3-ex^ + 3^ca^ + S'dx' + &c. 

 and COS. 4z = I — 4az + 4-B2' + 4^c«^ -f 4^ds' + &c. 

 Then taking the 1st differences we obtain, 



— 2 sin. -!■ z sin. 4 2 = — a* + ('i' — 1") bx^ + (2^ — 1^) cz^ + &c. 



— 2 sin. i s sin. ^z= — az + {3- — 2') bx" + (3^ — 2^) czl + &c. 



— 2 sin. i 2 sin. ^ a = — as + (4" — 3") bs' + (4^ — 3^) cs^ + &c. 

 Developing in series the factors of the 1st members, the )st terms of these 



members contain the 2d powers s' of s, and these members contain not the 1st 

 power of s. Therefore in the 2d members, the 2d power of s must also be 

 wanting; so that a = O. It is shown in the same manner, and according to 

 what is developed in ^ 12, that the odd powers of z are wanting in the ex- 

 pression of the cos. s ; so that the cosine is a function of s of the form 



COS. S = 1 — as' + BS^ + CS"' + DS' + &C. 



therefore cos. 2s = 1 — 2'as- + 2^bs^ + 2''cs'^ + 2''ds" + &c. 

 and COS. 3s = 1 — S'as'^ + 3^bs^ + 3^cs« + 3*'ds' -f- &c. 

 and COS. 4s = 1 — 4'as' + 4^bs* + 4^02" + 4'*ds^ + &c. 

 Taking here the 2d differences, we obtain, 



— 2' sin.^ -1- z cos. 2s = — A'- (3^ 1 ') As' + A" (3^ . .1*) Bs* + 8sc. 



— 2' sin." 4- s cos. 3s = — A" (4' ... 2') as' + A'i (4*. . .2*) Bs* + &c. 



— 2- sin.^ -L s COS. 4s = — A" (5'' ... 3") as" + A" (5'. .3') bs"* + &c. 

 Reducing into series the 1st members of all these equations, their 1st term is 



— z". But the first term of the 2d members is — 1.2. as\ Therefore 

 ] = 1 . 2 . A ; and a = - — -. 

 Taking successively the 3d and 4th diffs. we obtain, 



2" sin.* ^ z cos. 3s = a'" (4*. . .1^) b-^ + a'^ (4°. . .1*) cs" + &c. 



2* sin.* -1- s COS. 4s = Aiv (5\ . .2*) bs' + A^ (^56. . .2'') cs'' + &c. 



2* sin.* i s COS. 5s = A" (6*. . .3*) bs' + a^ (Q'^. . .a^) cs** + &c. 



1 



Whence in like manner, 1 = 1 . 2 . . . 4b ; and b = 



Taking likewise the 5th and 6th diffs. we obtain, 



— 2" sin.*^ 4- s COS. 4s = A" (6« l") cs*^ + av (6« 1 ^) ds' + &c. 



— 2" sin.*^ i s COS. 5s = A'i (7^ . . . 2") cs" + A"'' (7** . . . 2") ds'* + &c. 



— 2" sin.« i z COS. 6s = a^' (s'^ . . . 3") cs""' + a^' (8« . . . 3«) ds' + &c. 

 Hence — 1 = 1 . 2 ... 6c ; and c = — -— ^. Also d = + — ; and 



1 



^ "" 1 . 2 ... 10* 



Therefore cos. z = I — -^^ s' + ^ ./ ^ z' — -j— ^^ — 5. s* + &c. 



§ 14. Since sin. z = z — ~T73 "^ "^ TTTTI ^' ~ TTT.TT? ^' "*" ^'^• 

 And COS. z = i--^^z' + ^- z^ - j^^^ z' + &c. 



