TRANSACTIONS OF SECTION A. 431 
m? —2bm +d =0, 
mn+ nm—2bn—2cm+2e=0, 
n*— 2en+f=0. 
It hence easily appears that any given function of m,n can, by aid of the five 
parameters 4, c,d, e, f, be expressed in the form 4 + Bm+ Cn+ Dmn. 
This form containing 4 arbitrary constants, it follows that in general any given 
matrix of the second order can be expressed as a function of mand 7; for there 
will be four linear equations between A, B, C, D and the four elements of the given 
matrix. But this statement is subject to two cases of exception. 
The first of these is when m and m are functions of one another: for in this case 
A+Bm-+ Cn+Dmn is reducible to the form P+ Qm, and there will be only two 
disposable constants wherewith to satisfy the four linear equations. 
The second case is when the determinant of the fourth order formed by the 
elements of the four matrices | %, on pe ? ; 
respectively, it is not difficult to show that the value of this determinant is 
= (¢,7, —Tefs)? + {1 -t,) T2- (1-74) B} L(G - 4) T3—(T,—T,) ts}. 
This expression is a function of the five parameters 8, c, d, e, f, as may be shown 
in a variety of ways. 
Thus it is susceptible of easy proof that if w,, 4. are the roots of the equation 
p?—26u+d=0, and 1, v, the roots of the equation v?—2dy +f, then, the two ma- 
T15T2 
| vanishes; writing m,n= 
Ta p%q 
trices being related as above, we must have Ke he 1) Ge : ; Sp , and con- 
sequently, by virtue of the middle one of the three identities, p,», + 1,7, —2e=0. 
Writing this in the form 
fa. (41% + HoY2 — 2) (MyM + fav; — 2e) =0, 
this is 
Ae? —Qe, Abe + (uy? + pa”) (,? + 14") + 2 yptQ0,¥, =0, 
e? —2bee + b°f + Pd—df=0; F 
the function on the left hand is the invariant (discriminant) of the ternary quantic 
appurtenant to the corpus, and we have this invariant=0 as the necessary and 
sufficient condition of the involution of the elements of the corpus; the invariant in 
question is for this reason called the inyolutant. 
Expressing the values of the coefficients in terms of the elements of the two 
matrices, viz. 
which gives 
d=tt,—tgts, 2e=tyr, +74t,— tT — tT 0, f= T1T4—ToTgy 
it at once appears that the two expressions for the involutant are, to a numerical 
factor prés, identical. 
It can be shown @ priori that the involutant of a corpus of the second order 
must be expressible in terms of the coefficients of the function ; and therefore, being 
obviously invariantive in regard to linear substitutions impressed on m, n, it must 
be also invariantive for linear substitutions impressed on z, x, y, and must therefore 
be the invariant of the function. The corresponding theorem is not true, it should 
be observed, for the inyolutant of a corpus beyond the second order; for such inyo- 
lutant cannot in general be expressed in terms of the coefficients of the function. 
The expression for the involutant in terms of the ¢’s and r’s may also be ob- 
tained directly from the equation (m—p,) (n—v,)=0. To this end it is only 
necessary to single out any term of the matrix represented by the left-hand side of 
the equation and equate it to zero: the resulting equation rationalised will be found 
to reproduce the expression in question. 
I have thus indicated four methods of obtaining the involutant to a matrix- 
corpus of the second order; but there is yet a fifth, the simplest of all, and the 
most suggestive of the course to ke pursued in investigating the higher order of 
involutants. 
