»— I ir 



28 



, those of 



o/'Cotes's Theorem. 



Again, the values of ij; being i};, ;}; + 



I 1 " 



V + tj/, or 27r — ( TT — i[; ) will be 



, . nir . (w-f-i) w . , (271— i)ir 



II I 



Now, COS. 27r — ( TT — rj/) = COS. ( TT — ^). Hence, the product 



of all the denominators of p, p, &c. will be ( 1 — sx . cos. 4^ + 



I 2 I 



V)(l-2X.COS.(>H-f)+X-)....(l_2X.COS.(4. + <^)+X'). 



1 I ' 



COS.(4' + v)+V)....(l-2X.COS.(^+ll;^')+V)=, 



by CoTEs's formula 



1 — 2X^" . COS. 2« i|/ -|- x^, 

 I 



Thus we have, combining these separate processes, 



sin. n ( 1- 4') 



p.p....p = 2^ (i-e-) -(i+M •Ti::i;?r-,^n;rr+;?^ 



I z 



(i+X*)*". sin. « (—+>]/ (I— X')»".sin. nf— + 4,] 



H } 



-f * I — 2A^" ."cos. 2W^V-^-A♦" ""* * 1 — 2>." . COS. 2n^I/-j-X*" 



= 2a . 



. 1. Let n be any number of the form 4^ + 1 ; then 



,« (I+X*)*" . COS. «4/ fo 1 



I « I 2A»« . COS. 2m4/+X+" 



I 



If now tj/ = 0, or one of the p coincide with the transverse axis. 



Ip" (I+X*)*" 



p — 



n 



n ( I — V 



Let, next, ;]/ = - \n continuing = ^m + 1 1 and, 



P....'p=±/V^T.^ .... {8,4} 

 In the parabola, x = 1. Hence in this case. 



{8,3}. 



