508 
SOLUTIONS OF A SMITH’S PRIZE PAPER FOR 1871. 
[537 
a series of equations giving the value of the common root - (= a) in the several forms 
1 x dR dR 
x dR dR p 
my da ' db ’ m—ly db ' dc 
And it is clear that we have in like manner 
, , , „ dR dR 
x m : mx m ~ 1 y : &c. — -j-, : -jrr : &c. 
3 da db 
It is clear that if TJ involves, in any manner whatever, the coefficients a, b, ... 
which do not enter into the function V, then we have in precisely the same manner 
dU dU . dR dR s 
da : dF : &C ' = db : db : &C ” 
a system of equations satisfied by the values x, y which belong to the common root. 
But if the coefficients a, b, ... are contained in any manner whatever in both of 
the functions TJ, V; then by altering a, b,... we alter the common root; say that 
x + 8x, y + By belong to its new value; then we have 
dU. dU* ^dU dU, 7 ^ n 
& + Iky + s Ba + ^ 86 + ■ • • - °' 
dx 
db 
dV. ,dV~ ,dV d V „ , _ n 
—7— ox H—j— oy H—z— ba -|—jj— bb -h • • • — 0. 
dx dy da db 
Now the values of x, y which satisfy U = 0, V = 0 also satisfy 
dU clV_ 0 ' 
dx dy dy dx ’ 
hence from the foregoing two equations eliminating Sx or By, the other of these two 
quantities will disappear of itself, and we thus obtain an equation 
ABa + BBb + ... = 0, 
which must agree with the above equation 
or we have 
dR 
da 
Ba + 
dR 
db 
Bb + ... = 0, 
dR dR 
da ' db 
: &c. = A : B : &c., 
a system of equations satisfied by the values x, y which belong to the common root. 
In the case of a single equation TJ = 0 having a double root, the condition for 
this is A = 0, where A is the discriminant of the function TJ; and the like reasoning 
shows that for the values x, y which belong to the double root we have 
dTJ dU p __dA d A 
da db C ’ da ' db ' " ‘ ’