of Cotes's Theorem. 25 



TO-; tcr = t3- + — ;....t^=:7^ + illzill^ be written for ts, 



121 n I 



and for | a function of m, given by the equation 



cot. i == i±^ . tan. ^ -nr.] 

 be put 6 ; .... 5, the results of that substitution 



I n 



-« / '^ \" I-2'X'' . COS. MT^ + X"" 



COS. 6. COS. 6 . . . COS. = e^ .7') . ;; ^-7-^ • -J 9,1 f ' 



\ ' 1 — 2X" . COS. nzj-^K^' L^ 3 



I 2 « , 



If ts- = 0, whatever be w, 

 I 



f n n^ "s 2 



COS. e.,. COS. 5 = ^ . -| . U-^ =^ . -^; _ .|9,2|. 



If 7/ = (SW + 1) - 



' r — — • " 'i 

 cos.6....cos.5 = e .•'^— ^^ I9.3I. 



If/i = f2W + l) .-^, 



t 



COS. 9....C0S. fl = . .-^^ ...... [gA. 



If 7Z = (6m + 1) . 



3 , 



cos 5 . . . . COS. 5 = ^ 

 I n 



(f) 



If 7Z = (6;;z + 2) .-^ 



E~ir5 ) {^'^}* 



MDCCCXIir. 



