OF THE EOOTS OF SYSTEMS OF EQUATIONS. 
521 
§ 12. Symmetric Functions of Differences. 
69. In the unipartite theory there is a transformation which connects the 
Symmetric Functions of the Differences of the roots of the equation 
x”' — + n [n — 1) — • • • + (~)’* = 0 
with the non-unitary symmetric functions of the roots of the equation 
a:" — + Gbyx,”'~^ — , . , = 0. (^^ = oo ) 
In fact, the annihilating operator is in each case found to be 
9i — “h ^^2 + *^2 + . . . 
The theory of the Invariants and Covariants of a Binary quantic may be thus 
brought to depend upon non-unitary symmetric functions. {Vide ‘ American Journal 
of Mathematics,’ vol. 6, p. 131.) 
In the present case, there is also a transformation. For the purpose in hand, vuite 
the fundamental identity in the form 
(l + ccyc -j- (1 + ayr + (iff) . . . (1 + a„a: + jBi,y) 
711 
= 1 + na^QX -h na^yy + . - . + -p-ff + • 
Any function of the differences of the quantities on the left remains unaltered, when 
we write for the quantities a^, /3s, respectively li and ^s + h- The coefficient of 
xPy^ on the right then becomes 
S (a^ + h) {a^ li) . . . (o^j + li) (/3^ + i h) (Aj + 3 + /i) • • • {/dp + q + ii), 
which is 
(lOr 01 ?) -V {n ~ 2 ^ — ci^ \ ) {(lO?’"^ 01?) + (10?^ 01?-^)]/^ 
+ ~ g + - q + ^) f^(^iQp- 2 ^q^ q_ 2 dl?-i) + (10?’0T?--)]/i 
+ . . . 
But 
(To?’ 01?) = 
Hence is transformed into 
711 
(ti -p - q) ! 
ttpq- 
Id 
+ "b + ‘^^p-i,q-\ + 31 + 
the general term being 
Cp —s^ q — t ^ 
s\t\ 
3 X 
MDCCCXC.—A. 
