488 
ME. A. CAYLEY’S BLEMOIE 02^ CITEYES OE THE THIRD OEDEE. 
function on the left-hand side of the equation. The coefficient of in 11 is 
■ !■ /Y^ /Y^ 
fct'jwgwg 
-\-2l{c(^,xly^z^-\- &c.) 
-j-4:P(xfy,z,y,Zs+ &c.) 
-i-8l%y,y,z,z,z,; 
and it is easy to see that representing the function 
(1, 1, 1, yl—al, ctri—^iy 
(a, b, c, f, g, h, i, j, k, IJa, f3, y)\ 
the symmetrical functions can be expressed in terms of the quantities a. b, &c., and that 
the preceding value of the coefficient of in IT is 
a^ 
+2/(9hj-6al) 
+ 4^^(6gk- 3fj - 3hi+ 3P) 
+ 8^ffic ; 
and substituting for a, &c. their values, this becomes 
+ 4P{ - 6(rr+2/|;7^)(r;j-l-2Z|r) 
+3(;?t^+2^^:f)(r?+2zr^^) 
and reducing, we obtain for the coefficient of x'^ in 11 the following expressioi ,— 
-18/|Vr 
-24/Xp?T+rf+IV) 
Now the coefficient of x^ in F.U is simply F, which is equal to 
|6 _|_ ^6 4. ^6 _ 2^3^3 _ 2^3^3 _ 2 
-24/|Vr 
-24P(f+,j^+r)l^r 
-48/Wr; 
