of Cotes's Theorem, ii 



wherefore cos. a = ^— — and cos. a = ^-^ — . 



Thus, Xzsic" ""* and x'= r* ~', where ^ = i + — + rt 



' ■ 1 ' I.Z 



-j — ^ — |- &c. Hence, x" + ^"" = ^ • ^os. wa, and x'" + x'~" 

 = 2 . cos. wa';x — X =2v— 1. cosm. «a, X' — X = 

 2 V^ -- 1 . sin. nx'. Consequently, if k be any arc 



1 -« 2X'' . COS. ;^ + X^" = X" {x"+ X""" — 2 . cos. k} = ^x"" . 



sm. ( i-.).sin.(-^). 

 In like manner 



1— 2X' . cos. k + x' = 4X' . sm. (-^7—) • sin. ( — j- K 

 We will now proceed to the application of these equations, 



and first, in Equation j 1 1 for 6 substitute, successively, each 

 of a series of angles 



5; 5 + - = 6; 0+^ = 5; 9 + ^J!tzLlh = Q^ 



' ' n ' • n ' • n 



II 21 31 n 



And let the resulting values of r '^ be 



,C>;r«; /■> 



12 X 



we have then 



^ -^ ^ "" " • i-2X".cos. mG + a^" 1 '^ i 



12 n I 



for the product of the several denominators of the r^*^ will be 



j 1 — 2X.C0S. 5 + X'| |l— 2X .cos. 5 +>^*| ^ J 1— 2X. 



cos. + x"| = 1 — 2x" . COS. n^ -j- ^^" '^y Cotes'^ theorem. 



n I 



This equation appears under an imaginary form when e-ji, 



but, since cos. "* ^ is then a real angle, if we express it in a, 



C2 



