104 
ON LINEAR TRANSFORMATIONS. 
[14 
a, 
b, 
c, 
d, 
e, 
f, 
9 
b, 
c, 
d, 
e, 
f> 
9> 
h 
c, 
d, 
e, 
9> 
h, 
i 
which is true generally. 
Omitting for the present the theory of derivatives of the form 
□ JJVW = 23“ 31 ^ Ï2 v UVW, 
we pass on to the derivatives of the fourth degree, considering those forms in which 
all the differential coefficients are of the same order. We may write 
□ UVWX = (12.34)° (13 . $2/ (14.23) y UVWX = D a> p, y (U, V, W, X) = D a , Pty ; 
or if for shortness 
12.34 = M, 13.42 = 33, 14.23 = 0D, 
we have D a> p, y = f£l a 33^ QL y . UVWX. 
Suppose U= V = W — X, and consider the derivatives which correspond to the same 
value f of a + /3 + y. The question is to determine how many of these are independent, 
and to express the remaining ones in terms of these. Since the functions become 
equal after the differentiations, we are at liberty before the differentiations to inter 
change the symbolic numbers 1, 2, 3, 4 in any manner whatever. We have thus 
Da, /3,y = D^ v> a = D y> a, 0 - (~Y At, y> p — (~)- f D y> p> a — {-)- f D Pt 
but the identical equation 
& + 23 + = 0, 
multiplied by ^t°33 6 (ir c and applied to the product UVWX, gives 
D<x+i,b,c V D a b+\,c -h D a i) C+1 = 0, 
whence if a+b + c =f— 1, we have a set of equations between the derivatives D a P y 
for which a + 8 + y = f. Reducing these by the conditions first found, suppose ©/ is 
the number of divisions of an integer f into three parts, zero admissible, but permu 
tations of the same three parts rejected. The number of derivatives is 0/, and the 
number of relations between them is 0 (f— 1). Hence ©/—©(,/—1) of these derivatives 
are independent: only when f is even, one of these is Df 0 0 , i.e. 12‘ 34 / . UVWX, 
i.e. Vi!UV. 34 / TFX, or B f {U, V)Bf(X, W), i.e. [Bf{U, U)] 2 ; rejecting this, the 
number of independent derivatives, when f is even, is 0/—©(/—1)—1. Let 
be the greatest integer contained in the fraction -; the number required may be 
shown to be 
e{ or E-/-V, 
o o