Adams. — On Natural Sines. 



409 



Expanding the cosine and sine by Maclaurin's theorem, 

 we have the usual series, — 



cos z 



2~1 



!-*-,+ 



4 ! 



sin z — z 



+ ;n 



61 + 81 — &C> 



3 ! ' 5 ! 

 where z is expressed in radians. 



These series are not in convenient form for numerical 



cos 2 = 1 i- • A + -t • B « ' C + -* 



and sin z = — • a — - 



n it 



^ ~ 2 1 



calculation, so put z = — ■ h 



, . w 2 1 /ir\ 2 »t 4 1 /tt\ 4 m 6 1 /7r\« , . 



then cos* = l-^ • ^) + ^ • ri \j) -~ 6 ._^j+&c. 



and sin z = — . - _— - . — ( — 4. — . — ( — ) _ .w. 

 n 2 n 3 3!\2/^n 5 5!\2/ 



For convenience these may be wi-itten 



D -- &c. (P) 



6 + s- ■ c - - Sr ' ci + &c - (Q ) 



where (Callet, pages 27 and 28) — 



(~J = 1-23370, 05501, 36169, 82735, 43 

 (j)* = 0-25366, 95079, 01048, 01363, 66 

 (-0 b = 0-02086, 34807, 63352, 96087, 31 

 (y) 8 = 0-00091, 92602, 74839, 42658, 02 

 (|-) W = 000002, 52020, 42373, 06060, 55 

 (y) U = 0-00000, 04710, 87477, 88181, 72 

 (y)" = 0-00000, 00063, 86603, 08379, 19 

 (^y*= 000000, 00000, 65659, 63114, 98 

 (~y 8 = 0-00000, 00000, 00529, 44002, 01 

 (~Y°= 0-00000, 00000, 00003, 43773, 92 

 (-|-)*= 0-00000, 00000, 00000, 01835, 99 

 (-0 24 = 0-00000, 00000, 00000, 00008, 21 

 (-jY' 3 = 0-00000, 00000, 00000, 00000, 03 



B = 



c = 



D = 



E = 



F = 



G = 



H = 



I = 



J = 



1 



Ti 

 1 



6l 



1 



81 



1 



10T 



1 



12T 



1 



HI 



1 



16T 



1 



18~! 



1 



20~! 



TT 



-"■ — 22 ! 

 L =241 



M =26! 



&c, and 



