ON LINEAR TRANSFORMATIONS. 
112 
[14 
where the symbols £, g are supposed not to affect the x, y which enter into the 
expressions S. But we have identically 
an equation which gives rise to reductions similar to those which have been found for 
the derivatives D Ut y , but which require to be performed with care, in order to avoid 
inacccuracies with respect to the numerical factors. It may, however, be at once 
inferred, that the number of independent derivatives G a>l3)Y is the same with that of 
the independent derivatives D a> j3j y for the same value of a + /3 + 7. 
From similar reasonings to those by which B [U, B (U, U)) has been found, the 
following general theorem may be inferred. 
“ The derivative of any number of the derivatives of one or more functions, or 
even of any number of functions of these derivatives, is itself a derivative of the 
original functions.” 
For the complete reduction of these double derivatives, it would be sufficient, 
theoretically, to be able to reduce to the smallest number possible, the derivatives of 
any given degree • whatever. This has been done for the derivatives of the third degree 
Ca p -y, and for those of the fourth degree, in which all the differentiations rise to 
the same order (D„ )j3)y ): it seems, however, very difficult to extend these methods even 
to the next simplest cases,—extensive researches in the theory of the division of 
numbers would probably be necessary. Important results might be obtained by con 
necting the theory of hyperdeterminants with that of elimination, but I have not yet 
arrived at anything satisfactory upon this subject. I shall conclude with the remark, 
that it is very easy to find a series, or rather a series of series of hyperdeterminants 
of all degrees, viz. the determinants 
a, 6, c, d &c. 
b, c, d, e 
c, d, e, f 
d, e, f g 
a, b, c 
b, c, d 
c, d, e 
a, b 
b, c 
a, 3 6, 3c, d &c. 
b, 3c, 3d, e 
a, 4 b, 6c, 4 d, e &c. 
b, 4c, 6d, 4e, f 
c, 4 d, 6e, 4/, g 
a, 46, 6c, 4 d, e, 
b, 4c, Qd, 4e, /, . 
c, U, 6e, 4/, g, . 
а, 26, c 
б, 2c, d 
a, 36, 3c, d, 
. 6, 3c, 3d, 6, 
а, 36, 3c, d, 
б, 3c, 3d, 6, . 
а, 26, c, 
б, 2c, d, 
[I have inserted in these determinants the numerical coefficients which were by 
mistake omitted.] 
However, these functions are not all independent ; e.g. the last may be linearly 
expressed by the square of the second and the cube of (ae — 46d + 3c 2 ) ; nor do 
I know the symbolical form of these hyperdeterminant determinants.