ON LINEAR TRANSFORMATIONS. 
101 
14] 
Passing to the third degree, we may consider in particular the derivatives 
□ UVW = 23* 31* 12* UVW =G a (U, V, W): 
writing for shortness 
A = ^ s 2a-r£ r TJ — 77. r 
— r -y , Ox Oy U — U , 
we have the general term 
C a (U, V, W) = Z {{-) r+g+t A r A s A t U>*+ t -*V'«+r- t ‘W’* + *-r}, 
where r, s, t extend from 0 to a. By changing the suffixes r, s the following more 
convenient formula 
G a (U, V, TP) = SS{(-)^^pF-TT^-p-^[(-)^ p _^ (r+i _ a 4 t ]}, 
where t extends from 0 to 2a: p, cr, and 3a — p — a must be positive and not greater 
than 2a. 
In particular, according as a is odd or even, 
G a (U, U, U) = 0, 
C a (U, U, U) = QXX{(-y^U^U^-^X[(-yA p . t A <r+t . a A t ]}, 
omitting therein those values of p, a for which p > cr or a > 3a — p - a, and dividing by 
two the terms in which p = a or cr = 3a — p — cr, and by six the term for which 
p = cr — 3a — p — cr, = a. 
In particular, for functions of the fourth or eighth orders we have the constant 
derivatives 
ace — ad 2 — b*e — c 3 + 2bed,; 
aei — 4ibd — 4afh 4- Sag 2 + 3tc 2 + 12beh — 8chd — Sbgf— 22ceg + 24c/ 2 + 24d 2 g — 36def+ 15e 3 ; 
the first of which is a simple determinant. Thus we have been led to the functions 
ae — 4bd + 3c 2 and ace — ad 2 — eb 2 — c 3 + 2bcd, which occur in my “ Note sur quelques 
formules &c.” (Grelle, vol. xxix. [1845] [15]), and in the forms which M. Eisenstein has 
given for the solutions of equations of the first four degrees. 
Let U be a function of the order 4a: the derivative G may be expressed by 
means of the derivatives B. 
For, consider the function 
B U [U, B n (V. V)]-, 
paying attention to the signification of B, this may be written 
id ia 23 2a UVW,