OF THE ROOTS OF SYSTEMS OF EQUATIONS. 
403 
18. Bj comparison of these relations with the corresponding algebraic ones to 
which reference has been made, it is manifest that and ai’e respectively in co¬ 
relation with Spq and a^rj- In other words these operations respectively correspond to 
the partitions [pij) and OB). It is necessary to find the operations which 
correspond to the remaining partitions which symbolize symmetric functions. 
We have the easily derivable results in operations 
9pi'ii9p/h — 9px'u9pi'h “f" 9pi+Pi, 
cj _ 2 I 
9pxq^ — 9pi'h “T 99.Pi, 9qi‘> 
9Piqi9Pi'h9PA ~ 9 pa 9p/h9PA 4 " 9pa 9Pi+Pi,q-2 + qz 4 “ 9pa 9pz+pi.qz-^qi 
4 " 9pzqz9Pi+ih, ?i + 5'2 4 ~ 9pi+P2+ih,qi + qz + q-p 
9PA 9PA — 9PA 9PA 4 “ ‘^9pa 9Pi+Pi, qi + q-i 4 - 99Pi,9qi9 pa 4 ' 9iPi^P2,'2qi + q 2 ^ 
9PA — 9PA 4" '^99Pi,9qi9pA 4" 9zPi, 
37i ’ 
where as usual the bar denotes symbolic multiplication ; and comparing these wuth 
the algebraic formulse 
(]Mi) imi) = (Pfiii 2Mz) 4- (2d 4- 2d, <li 4- q-z) 
{Whf = 2 -f 2q,) 
(Jd'Zi) {ihq-i) il¥h) = {PiqiPzq-zlhqs) 4- {ihqi Ih 4- 2d, 99 4- 9?) 4- {pphlh 4- J>i, 93 4" 9\) 
4- ilMz Pi 4" Pi, 9i 4- ^ 3 ) 4- {p\ 4- Pi 4- id, 9i 4- 92 4- is) 
(mif (m 2 ) = 2 im? Sd) + 2 (2dii id 4- id, 9i 4- i3) + (2id, ‘^9i id'd) 
4- ( 2^1 4 - Pi, 2ry^ -f f/g) 
(P\9(f‘ = 6 {Pi9i) 4- 3 {2pi, 2(/i pqp) 4- (3pi, 3vi) 
it is evident that the operations 
^JpA , 
2 ! 
•dv/i ^ihA 
2 ' 
JpA 
a I 
are produced according to the same Ioap as the symmetric functions 
(idii^), (pi9iPPi()^ (ppii ); 
further the law is perfectly general and indicates that the operation 
(t(\ (rP. ’ • • 9pa"9pa' • • • ’ 
