of Cotes' s Theorem. 17 



3» _3«\ 



4- ^ r ) — a . I~3r\ • • • • J 4>5 [ 



The Equation |5| maybe treated in the same manner, for 

 the values of tr, being w\ t?r + — ; ts- + ^^"~ , those of 



tr — 3- in an inverted order will be (ifi; = — {^3- + ^"~^ |) 



I 



, Ztt , 2 C« — i) T 



"> " + T' " + -^ • 



Hence we find ^ 



(2) (2) n I — zx" . COS. Mj+x*" 



' ^ " • (i+x^)" * 



or, since cos. v = cos. « ( tt + ^ ) 



I 



, . , . I — 2\" . COS. n Cw+w) +x*« 



, ^ —'' • (i+A^)" • • • 1^'M- 



As this case, however, is manifestly similar to that of {4,1 1, 

 we shall pursue it no farther. 



The 6th of the equations, in page 9, offers, however, some 

 results worthy of consideration. By treating it like the rest, 

 it becomes 



/*^ /'^ I — 2X'' . COS. nz7 + x*" 



• n I S f\ l\ 



^(2) ^2) I— 2X" . COS. n(7r + ^) +X*" • • • • (. ' J 



I a 



1 . When ta- = 0, it becomes 

 I 



MDCCCXIII. D 



