OF THE ROOTS OF SYSTEMS OF EQUATION'S. 
489 
aiiclj iu general, becomes converted into 
d” H'^p—hq + VClp^ ^ _ 2^, 
Hence any rational integi’al function of the coefficients 
CtjO, CQi, rtoo> 0-025 • • • 5 
J {ciiQ, «oi> *^005 « 02 , . . .)z=J, 
VIZ., 
is converted into 
wher 
•e 
J + + ^i/0l)/+ o', + ^90lYf'^ or • • • > 
9iq — '^'-^p-i,q ’ 9oi — ^"p'l ’ 
and the multiplication of operators is symbolic. 
The new value of J‘ is 
exp (ppio + 
where the bar is placed over exp to denote that the multiplication of operators is 
symbolic {vide Art. 7). 
13. Write 
1 - 
f^ \ 9\d‘ i/offi 
the bar denoting symbolic multiplication, then 
exp (pyio + 
— (1 d~ ffi H~ pvGji + + • • • + + . . .)/. 
(Compare Art. 7.) 
Now suppose the symmetric function f expressed in terms of 
to be 
^l> /^l> ^2’ '^3> ^3’ • ■ • 
{ Pdh P2<l2 lh% • • •)• 
The introduction of the new quantities p, v results in the addition to 
{pph P2d2 Pdh • • •) 
of the terms 
(^V727^353 • • •) + {PdhliMz •••)-!- P^'^d^^iPiPPdh 
* By “symbolic” is to be understood “ uon-02oerational,” as iu what is commonly known as the 
“ symbolic ” form of Taylor’s Theorem. 
MDCCCXC.—A. 3 It 
