14] 
ON LINEAR TRANSFORMATIONS. 
109 
We may now proceed to demonstrate an important property of the derivatives of 
the fourth degree, analogous to the one which exists for the third degree. Let 
If, V, W, X be functions of any order f: then, investigating the value of the expression 
B 2f _ 2a [B a (U } V), B a {W,X)\ 
this reduces itself in the first place to 
e$ f -~ a 12“ 34“ UVWX, 
where rjo refer to U and V, and |^, to W and X: this comes to writing 
& = £i + £>, Ve = Vi + V2> an d £<*> = £3+^4, V<t> = Vs + Vi', whence 
tty = 13+ 11+23+ 24, 
or the function in question is 
(13 + 14 + 23 + 24) 2/_ ‘ a 12“ 34“ UVWX. 
But all the terms of this where the sum of the indices of % 1 , or %. 2 , rj 2 or 
V3 or ^4, Vi> exceed f, vanish : whence it is only necessary to consider those of the 
form 
K r (13.42/ (14.23) /_ “~ r (12.34)“ UVWX, 
where K r denotes the numerical coefficient 
(-Y [2/- 2a]tf~ 2a 
[r]'' [r] r [/— a — ry~ a ~ r [f— ol — rY~ a ~ r ’ 
or Bof-ta [B a (U, V), B a (W, X)] = X {.K r D a , r , (U, V, W, X)}. 
In particular, if U = V= W = X, writing also B a for B a (U, U), 
B 2 /-2a(B a , B a ) = X (K r D a r j_ a _ r ). 
If a is odd, this becomes 
0 = X (K r D a r i y_ a _ r ), 
an equation which must be satisfied identically by the relations that exist between 
the quantities D: if, on the contrary, a is even, we see that there are as many 
independent functions of the form 
B 2 f-2a{B a , Ba) 
as there are of the form D; and that these two systems may be linearly expressed, 
either by means of the other. Thus, for the orders 3, 5, 7, the derivatives D are 
respectively equal, neglecting a numerical factor, to 
Byu\ IP), B W {U\ IP), Bu(U\ U 2 ); 
for the sixth order they may be linearly expressed by means of 
B 12 (U>, IP), B 6 2 , 
and so on. All that remains to complete the theory of the fourth degree is to find 
the general solution of this system of equations, as also of the system connecting the 
derivatives D.