13] 
ON THE THEORY OF LINEAR TRANSFORMATIONS. 
93 
and thence v s = Mv\ 
which coincides with a previous formula, and 
0U=MOU: 
whence, eliminating M, 
eu^ejj 
V 3 V 3 
an equation which is remarkable as containing only the constants of U and U: it is 
an equation of condition which must exist among the constants of U in order that 
this function may be derivable by linear substitutions from U. 
In the symmetrical case, or where 
U = ax A + 4 ¡3cc?y 4- 6y x 2 y 2 + 4<8xy 3 + ey 4 , 
it has been already seen that v is given by 
v—ae — 4/38 + 3y 2 . 
Proceeding to form 6U, we have 
gl = a 3 e 3 - 4/3 3 8 3 + 3y 6 , 
33 = 4 (ae/3 2 8 2 - a 2 e 2 /38 + 3y 2 /3 2 8 2 - 3/38^), 
QL = 3 (a 2 e 2 y 2 + 2y 6 + aey 4 — 4/3 3 8 3 ), 
H) = 6 (ae/3-8 2 — 2ae/38y 2 + 3/3 2 y i 8 2 — 2(38^), 
(¡£ = 3ae7 4 — 4/3 3 S 3 + y*, 
Jp = 6 (a 2 8 2 ye + e 2 /3‘ 2 ya — 2fi 3 ey8 — 28 s a/3y — 4/3 2 y 2 8 2 . + 4 . /38y* + y 3 /3 2 e + y 3 a8 2 ), 
(5 = 12 (ae/387 2 — a/3yS 3 — ey8/3 3 + 7 3 a8 2 + y 3 e/3 2 + /38y 4 — 2/3 2 7 2 8 2 ), 
= (a 2 8 4 + e 2 /3 4 — 4/3 2 ey 3 — 4ay 3 8 2 + 6/3 2 y 2 8 2 ), 
and these values give 
6U= a 3 6 3 - 6a/3 2 8 2 e - 12a 2 ¡38e 2 - 18a 2 7 2 e 2 - 27a 2 8 4 - 27/3 4 e 2 + 36/3y8 2 + 54 a 2 y8 2 e + 54 a/3 2 ye 2 
- 54a 7 3 8 2 - 54/3 2 7 3 e - 64/3 3 8 3 + 81ay 4 e + 108a/3yS 3 + 108/3 3 y8e - 180a/3 7 2 8e, 
so that this function, divided by (ae — 4/38 + 37 2 ) 3 , is invariable for all functions of the 
fourth order which can be deduced one from the other by linear substitutions. The 
function ae — 4/38 + 3y 2 occurs in other investigations: I have met with it in a problem 
relating to a homogeneous function of two variables, of any order whatever, a, /3, 7, 8, e 
signifying the fourth differential coefficients of the function. But this is only remotely 
connected with the present subject. 
Since writing the above, Mr Boole has pointed out to me that in the transform 
ation of a function of the fourth order of the form aa1 4 + 4ba?y + 6ca? 2 y 2 + 4dxy 3 + ey 4 ,— 
besides his function 0u, and my quadratic function ae — 4>bd + 3c 2 ,—there exists a function