ON THE THEORY OF LINEAR TRANSFORMATIONS. 
94 
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of the third order ace — b 2 e — ad 2 — c 3 + Zbdc, possessing precisely the same characteristic 
property, and that, moreover, the function Qu may be reduced to the form 
{ae — 4bd + 3c 2 ) 3 — 27 (ace — ad 2 — eb 2 — c 3 + 2bdc) 2 ; 
the latter part of which was verified by trial; the former he has demonstrated in a 
manner which, though very elegant, does not appear to be the most direct which the 
theorem admits of. In fact, it may be obtained by a method just hinted at by 
Mr Boole, in his earliest paper on the subject, Mathematical Journal, vol. it. p. 70. 
The equations d x 2 u = 0, d x d y u = 0, d y 2 u = 0, imply the corresponding equations for the 
transformed function: from these equations we might obtain two relations between the 
coefficients, which, in the case of a function of the fourth order, are of the orders 3 
and 4 respectively: these imply the corresponding relations between the coefficients of 
the transformed function. Let H=0, B— 0, A'— 0, B'= 0, represent these equations; 
then, since A =0, B = 0, imply A'= 0, we must have A'= AA'+ MB, A, M, being 
functions of X, A', fM, &c. /A: but B being of the fourth order, while A, A' are only of 
the third order in the coefficients of u, it is evident that the term MB must disappear, 
or that the equation is of the form A'= AA. The function A is obviously the function 
which, equated to zero, would be the result of the elimination of x 2 , xy, y 2 , considered 
as independent quantities from the equations ax 2 + 2bxy + cy 2 = 0, bx 2 + 2cxy + dy 2 = 0, 
cx 2 + 2dxy + ey 2 = 0, viz. the function given above. Hence the two functions on which 
the linear transformation of functions of the fourth order ultimately depend are the 
very simple ones 
ae — 4bd + 3c 2 , ace — ad 2 - eb 2 — c 3 + 2bdc, 
the function of the sixth order being merely a derivative from these. The above 
method may easily be extended: thus for instance, in the transformation of functions of 
any even order, I am in possession of several of the transforming functions; that of the 
fourth order, for functions of the sixth order, I have actually expanded: but it does not 
appear to contain the complete theory. Again, in the particular case of homogeneous 
functions of two variables, the transforming functions may be expressed as symmetrical 
functions of the roots of the equation u = 0, which gives rise to an entirely distinct 
theory. This, however, I have not as yet developed sufficiently for publication. There 
does not appear to be anything very directly analogous to the subject of this note, in 
my general theory: if this be so, it proves the absolute necessity of a distinct investi 
gation for the present case, that which I have denominated the symmetrical one.