276 



ROYAL SOCIETY OF CANADA 



A Pi 



P. 1 



P, 



P» Pi 



P-. 1 



P. 



?5 



2p, 



Pi 

 1 



2p, 



Pi 

 1 







+ terms containing no p with subscript greater than 

 three, 



/>3 



P2 



Pi 

 ] 



+ terms containing no p with subscript greater than 

 four, 



Pu P, 2^2 P3 2p, ;)5 2p, 

 Pi, 1 /Ji P.. 2;?3 p4 2;)5 



?12 



Pn ^> 



1 

 





 



Fa Ps ^Pi 



Pi P2 PZ 



1 Pi 



1 







P2 



Pi 

 1 



+ terms containing no f with subscript 

 greater than seven. 



From the law exhibited here we may write for 2 a^a^ . . 

 Pi, Pi ^2 P3 2^4 Ps 2^6 Pi 



Pi-, 1 



Pu 



?13 



Pl2 f> 



Pu *» 



p« 



Pl P2 '^Pi Pi ^Pi Ps 



P, Pa 2;)4 p^ 



Pi jt>2 Pa ;'* 



1 Pi i'. Pa 



1 Pi P2 



1 ;>! 



1 



+ terms containing no p v^ath sub- 

 script greater than eight. 



8. The coefficients of ;?, and p^ in 2 a, a, . . . . a^ may now be 

 found by the method of undetermined coefficients. Assuming 

 coefficients for these terms and operating upon 2 «j «j . . . . a^ 

 by P we must get p^2 a^a^ . . . . «,. Equating the coefficients of 

 these terms involving the assumed coefficients we find their value. 

 If we denote the determinant in "2 «, «j . . . a^ by D^ we get in this 



way 2 tti a^ a^ = D^ 



+ Ps ip\ Pi - p\ P* + 2 ;>' ;^5 + 5 v\ Pe + 3 p, p, - Ps - 5 p, Pj p, 

 + 3pî P2 P« - 4 p, Pi Ps - S p, Pi P, + 5 P, Pî Pa - 5 Pz Pa + 5 p, p", 

 + 3 pI + 5 P3P5 - 18 PaPj - P; (2^ P5 - 2 p' p, + 6 p; p, - pÎ Pa p, 

 + pÎ Pa P. - 3 p] P2 P^ - Pi Pi Ps + ^ pI P5 - 5 P2 P, P« + 3 P3 p. 



