428 
[74 
74. 
ON THE SIMULTANEOUS TRANSFORMATION OF TWO HOMOGE 
NEOUS FUNCTIONS OF THE SECOND ORDER. 
[From the Cambridge and Dublin Mathematical Journal, vol. IV. (1849), pp. 47—50.] 
The theory of the simultaneous transformation by linear substitutions of two 
homogeneous functions of the second order has been developed by Jacobi in the memoir 
“ De binis quibuslibet functionibus &c., Grelle, t. xn. [1834], p. 1; but the simplest 
method of treating the problem is the one derived from Mr Boole’s Theory of Linear 
Transformations, combined with the remark in his “ Notes on Linear Transformations,” 
in the Cambridge Mathematical Journal, vol. IY. [1845], p. 167. As I shall have occasion 
to refer to the results of this theory in the second part of my paper “ On the Attraction 
of Ellipsoids,” in the present number of the Journal [75], I take this opportunity of 
developing the formula in question; considering for greater convenience the case of three 
variables only, 
Suppose that by a linear transformation, 
x = a x x + /3 y x + 7 ¿i, 
y = a! x x + /3' 2/4 + 7' z lt 
Z = Ct"x 1 + /3 "y 1 + 7 "z x , 
we have identically, 
ax 2 + by 2 + cz~ + 2fyz + 2gxz + 2hxy = a x x^ 4- bgjf + c x z? + 2f x y x z x + 2g x z x x x + 2h x x x y x> 
Ax 2 + By 1 -t- Cz 2 + 2Fyz + 202x + 2Hxy = A x xJ + Bgy? + C x z x 2 + 2F x y x z x + 2G x z x x x + 2J[- i x 1 y 1 . 
Of course, whatever be the values of a, b, c, f, g, h, the same transformation gives 
ax? + b y 2 + cz 2 + 2fyz + 2gzx + 2h xy = a^ 2 + b^ 2 + c x z x 2 + 2 i x y x z x + 2g x z x x x + 2h 1 (r 1 y ] ,