354 
EEY. E. HAELEY ON TIIE METHOD OF STMMETEIC PEODTJCTS. 
25. But a more complete verification is obtained by taking the sum of all the 
numerical coefficients that enter into the expression for 11. That sum is 
-4761104, 
as it ought to be. For the roots of 
are 
( 1 , 1 , 1 , 1 , 1 , l ^ r , 1)'^=0 
— 1, a, — a, ak — ak 
a being an unreal cube root of unity ; and for this particular form we have 
^J, = ll-10a, ^3= 21+lOa, ^3=1, 
0,= -29 , - 44-20«, ^^6= -24— 20a; 
CA=oSl, ^A=-29, ^)A=496. 
4761104. 
26. Another convenient verification is afforded by writing 
a, bh, 10^?, lOfZ, 6(?, f 
a, b, c, d, e, f 
respectively, when the sum of the numerical coefficients of the several powers oif should 
be zero*. And the transformed result will be worth having for its own sake, as belong- 
ing to Mr. Cayley’s standard 
(a, h, c, d, e, fjx, l)h 
* Not only the sum of all the numerical coefficients, but the sum of the numerical coefficients of each 
poieer off, is zero. The reason is, because the roots of 
(1, 1, 1, 1, l,fjx, 1)'’=0 
are included in the form 
or 
Consequently 
for 
— l + /, 
and therefore /3=0. So that in this case the symmetric product vanishes identically. 
In relation to the foregoing property, Mr. CiyuET remarks as follows: — More generally the forms 
(1, 0, 0, 0, 0, 1+fl^, y)\ (1, 1, 1, 1, 1,/XA fj 
are eqiuvalent, the modulus of substitution being unity, as is at once seen by writing 
y= y'; 
and the leading coefficients of any covariant of the same two forms respectively are therefore absolutely 
identical. That is, any seminvariant of the form 
{a, I, c, d, e,fyx, y)® 
will have the same value, whether we write therein 
{a, h, c, d, e,/) = (l, 0, 0, 0, 0, 1+/), 
or 
{a,l,c,d,e,f) = {\,\,l,l,\, f ); 
whence in particular a seminvariant which vanishes upon writing therein {f, c, d, e) = (0, 0, 0, 0) wiU also 
vanish upon writing therein {a, h, c, d, e) = (1, 1, 1, 1, 1) ; that is, in such a seminvariant the sum of the 
numerical coefficients of each, potoer of /is zero. 
