100 
ON LINEAR TRANSFORMATIONS. 
[14 
The above functions may be transformed by means of the identical equation 
B.(V, wo=irt*(F, W), 
to make use of which, it is only necessary to remark the general formula 
^i K Vi lx Z/v/Bk (V* W) = B k (l*nf V, IF). 
Thus, if k = 1, we obtain for the above series, the new forms 
ac — b 2 , 
(ae — bd) — 3 (bd — c 2 ), 
(ag — bf) — 5 (bf — ce) + 10 (ce - d 2 ), 
(ai - bh) — 7 (bh — eg) + 21 (eg — df) — 35 (df— e 2 ), 
&c., 
the law of which is evident. This shows also that these functions may be linearly 
expressed by means of the series of determinants 
| a, b 
b, c j 
We may also immediately deduce from them the derivatives B which relate to two 
functions. For example, for functions of the sixth order this is 
ag' + a'g — 6 (bf' + b'f) + 15 (ce' + e'e) — 20 dd', 
which has an obvious connection with 
ag — 66/+ 15ce — 10iZ 2 ; 
and the same is the case for functions of any order. 
The following theorem is easily verified; but I am unacquainted with the general 
theory to which it belongs. 
“If TJ, V are any functions of the second order, and W = \U + gV\ then 
W), B, ( W, IF)] = 0 
(where B 2 ' relates to g) is the same that would be obtained by the elimination of 
x, y between U= 0, V = 0.” (See Note 1 .) 
In fact this becomes 
4 (ac — b 2 ) (a'c' — b' 2 ) — (ac' + a'c — 2bb') 2 — 0, 
which is one of the forms under which the result of the elimination of the variables 
from two quadratic equations may be written. This is a result for which I am 
indebted to Mr Boole. 
j a, b, c &c. 
! b, c, d 
1 Not given with the present paper.