[ '58 ] 



^ is odd and — when even : alfo the produdl of all the values 

 ■when n is odd = „_, and when even = + 



n n 



n.— — I. — 



2 2 



« 



-2 - 



(to —terms) 



I. 2. 



— ■ 2 

 2 



I 

 n — I 



Whence the value of P O* x P O'* X &c. above found becomes 



n .^^ n — 2 ji_ n^—'l . " — 4 " 



«■■ + I — nx'^. ic^+i) + X* X xM-i" - +2X 



or expanding thefe terms 



I. 2 I. 2. .^ 



+ nx'^ + I 



«— 2 



2 n — 4 



— « .V^ X ^' + 1 = — « ;v — ff. « 2 .X — 



2 n — 6 



1 -v 



«X 



n— 4 



j«— 4 



«. K 2. n 3 



I. 2 



i-6 



i. 2 I. 2 1-2 



I. 2. 3 I. 2. 3 



&c. &c. * 



Hence 



* Mr.Simnfon in liis ElTays, page 115, has arrived by a different procefs at a fimilar 

 conclufion, and alTerts without any demoiiftration that the co-efficients deftroy each 

 other. Tliis however is tlie only dilhcalt flep in the whole propofition. 



