14] 
ON LINEAR TRANSFORMATIONS. 
99 
and it will be assumed that [JU only contains the differential coefficients of the 
/ th order. In this case, the derivative is said to be of the degree p and of the 
order /. The most convenient classification is by degrees, rather than by orders. 
Commencing with the simplest case, that of functions of the second order (and 
writing F, W instead of V 1 , F 2 ), we have 
OVW=12 a VW, 
(where , ip apply to F and y 2 to If). This will be constantly represented in 
the sequel by the notation 
12 a VW=B a (V, W). 
Hence, writing 
we have 
and in particular, according as a is odd or even, 
Ba(V, V) =0, 
continued to the term which contains F*“F ia , the coefficient of this last term 
being divided by two. 
Thus, for the functions \ (<ax? + 2bxy 4- cy 2 ), ^ (ax* + 4bx 3 y + 6cx 2 y 2 + 4dxy 3 + ey 4 ), &c., 
if a be made equal to 2, 4, &c. respectively, we have the constant derivatives 
a с — Ъ-, 
ae — 46d + 3c 2 , 
ag — 6bf+ 15 ce — 10d 2 , 
ai — 8 Ыг + 28 eg — 50 df + 35e 2 , 
which have all of them the property of remaining unaltered, a un facteur pres, when 
the variables are transformed by means of x = \x + gy, y = \'x + gy. Thus, for instance, 
if these equations give 
ax 2 + 2 bxy + cy 2 = àx? + 2bxy + cy 2 , 
àc — b 2 = (Ay/ — A'y) 2 • (ac — b 2 ), 
then 
and so on. This is the general property, which we call to mind for the case of these 
constant derivatives. 
13—2