576 
ME. A. CAYLEY ON TSCHIENHATJSEN’S TEANSEOEMATION. 
which for shortness may be represented by 
or say 
IC=( 
a. 
h, 
g. 
1 
h, 
b, 
f, 
m 
g. 
f , 
c, 
n 
1, 
m. 
n, 
P 
IB, C,D,E^), 
fC=(niB, C, D, 
Now, by a formula in my memoir “ On the Automorphic Linear Transformation of a 
Bipartite Quadric Function*,” if T denote any skew symmetric matrix of the order 4, 
then if 
(B, c, D, E)=(a-'(a-T)(a+T)-’niB', a, d', e'}, 
in which formula Q— T, (Q+T)“’, Cl are all matrices which are to be com- 
pounded together into a single matrix, we have identically 
(QIB, C, D, E)^=(QIB', C', D', E')^ 
Let Q denote the determinant |n-l-T|, then the terms of the matrix are 
respectively divided by Q, and we may write 
(n+T)-'=g.Q(n+T)-, 
where Q(Q-|-T)~' is the matrix obtained from the matrix (Q+T)“^ by multiplying 
each term by Q, the terms of Q(n-i-T)“* being thus rational and integral functions of 
the terms of the matrix (n+T). Hence if, instead of the before-mentioned relation 
between (B, C, D, E) and (B', C', D', E'), we assume 
(B, c, D, E)=(n-’(n-T)Q(a+T)-'niB', c', h, H), 
we find 
(QIB, C, D, E)^=QXniB', G, D', E^. 
And if the matrix T is such that we have Q=0, e. Det. (ri-j-T)=0 (which is a 
quadi’ic relation between the terms of the skew matrix, that is, each term is contained 
therein in the first and second powers only), then the equation becomes 
(QIB, C, D, E)^=0. 
It is clear that this can only be the case in consequence of the coefficients of trans- 
formation in the equation 
(B, C, D, E)=(Q-'(Q-T)Q(Q-1-T)-'QIB', C', D', E') 
being such that there shall exist at least two linear relations between the quantities 
(B, C, D, E), and I assume (without stopping to prove it) that they are such that the 
number of such linear relations is in fact two. That is, the last-mentioned equation 
estabhshes between the quantities (B, C, D, E) two Hnear relations, in virtue whereof 
* Philosophical Transactions, vol. eslviii. (1858), see p. 44. 
