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504 REPORT—1862. 
is changed into the product of two forms F.(y,, y,, .. yn) and F,(z,, z,,...z,) 
of the same order, and containing the same number of indletermitiates: Fr is 
said to be transformable into the product of F, and F,; and, in particular, if 
the determinants of the matrix 
| tasa.y |» 
which is of the type x n’, be relatively prime, F, is said to be compounded of 
F,andF,. Adopting this “definition, we may enunciate the theorem—*« If F, 
be transformable into F,x F,, and Ai F,, G,, G, be contained in G,, F,, F, 
respectively, G, is transformable into G, ibfce ss ‘and, in particular, if F be 
compounded of ', and F,, and the forms FE: Gg Ge be equivalent to the forms 
G,, F,, F, respectively, é, is compounded of G. and ee 
It is only i in certain cases that the multiplication of two forms gives rise to 
a third form, transformable into their product. Supposing that F, and F, are 
irreducible forms, 2. e. that neither of them is resoluble into rational factors, 
let I,, L,, I, be any corresponding invariants of F,, F,, F,, and let us represent 
by B and C the determinants 
dx, | = Troha fe 
dyg| B=1,2,3,...n,} 
and 
dx, o=1,'2, 3,75’. a 
dz, nam A, Wy ype Ms it 
The transformation of F, into F, x F, then gives rise to the relations 
I,xB* =I,xF,' 
I,xC*=1,x Fi, 
7 denoting the order of the invariants I,, I,, I,. If one of the two numbers 
I, and I, be different from zero, we infer that. m isa divisor of n. For if 
“ be the fraction = reduced to its lowest terms, the equations 
v 
I’x BY =I’ x F." 
Tok Or oF oe BS 
imply that F, and F, (cleared of the greatest numerical divisors of all their 
terms) are perfect powers of the order pu; 7. ¢., w=1, or m divides n, since F, 
and F, are by hypothesis irreducible. We thus obtain the theorem (which 
however applies only to irreducible forms having at least one invariant 
different from zero)—‘‘ No form can be transformed into the product of two 
forms of the same sort, unless the number of its indeterminates is a multiple 
of its order.” For example, there is no theory of composition for any binary 
forms, except quadratic forms, nor for any quadratic forms of an uneven 
number of indeterminates. 
Again, when m is a divisor of n, let n=km, and let 4, ¢, d,, d, represent 
the greatest numerical divisors of B, a a respectively; we find 
r=( a) = r)= a), Ba (Fs), O_(F 
T=( b wy Vey 8 Ne. 6 o=(B) . 
The first two of these equations show that the invariants of the three forms 
F,, F,, F, are so related to one another, that we may imagine them to have 
