the diversity of form arising only from the different arrange- 

 nient of the terms in the ^ powers of the multinomials. 



M 



Demonstration. The y powers of tlie expressions de- 

 duced from the Lemma (the co-efficients being understood) 

 Avill be of this form, viz. 



+QPr-'+RPF VSP^ +TP7 +VPr ' -f&c. 



QP?~'+QRP?~'+(QS+R)P?"'+(QT+RS)P^~'+&c. 



+QP~^ + QRP^~^ +(QS4.QR)P?~'+&c. 



+ QPr +QRPr * -H&c. 



5- m K 



+ QPf- +&C. 



H-DP'^"' +&C. 



n 



+ (QD ^_,+RD ^_„ +SD^_3 +&c.)xP?~" +&C. 



+ (QD„_2+Q^^«-3 + ^^I^»-4 +&c.)xP"-' +&C. 



+ (Qb„_3+QRD„_, ■'.- QSD ,_, + QRb _ .3+&C.) xP"-*&c. 

 + (QD^_^4-QRD^_5 + &c.) xPr +&C. 



By prefixing to the powers of the letters Q, R, S, &c. the 

 same powers of their variable co-efficients, or of the triangular 



numbers 



