288 
Proceedings of the Royal Society of Edinburgh. [Sess. 
form Cx^x?/ . . . x e n n , where e 1 + e 2 + . . . + e n = n and C is an integral 
function of a’s, — a result which Hesse writes 
eie 2 . . . «ri i 
X e JX e J 
X e n 
n > 
the coefficient of any term being denoted by an A with n suffixes identical 
with the n exponents of the cc’s. Now let us suppose the substitution 
to be orthogonal, in which case we know that 
j k=n 
Xjc = aifc£l + a 2/t&+ • • • + a nk£n > i 
) *=1 
and let us thereby transform en x [^ x ^ • • • X T so as t< o have it again 
in terms of the i’s. In doing this Hesse pays attention only to the term 
in £^ 2 . . . in, making the assertion that the coefficient of . . . £ n in 
xjix® 2 . . . x® n is either the same as the coefficient of xpxf 2 . . . x® n in 
... in or differs from the latter coefficient by a merely arithmetical 
multiplier. From this it follows that the coefficient of i^ 2 . . . i n in any 
term A eie2 . . . e n x l lx ? • • - X T is a merely arithmetical multiple of AJ iea>><en ; 
and, if the multiplier in question be denoted by © eiC2 , _ en , there results 
from the equatement of coefficients 
A 2 
en e x e 2 . . . en 
Next, let us suppose in addition that our substitution transforms an 
n - ary quadric 
ffx^x.2,. . . , x n ) into g£ x + + . . . 
a step which, as we know, introduces the quantities whose reality is in 
question. In regard to them Hesse first recalls Jacobi’s proof (1833) that 
they are such that 
+ • ■ • + gUl , giii+gH 1+ • 
9n£n > 
are also expressible as homogeneous quadric functions of the cc’s, and that 
the coefficients of these quadrics are rational integral functions of the co- 
efficients of the original quadric f v It is seen to be not inappropriate 
therefore to use 
fp( X l 1 3 
. . , x n ) for glil + gUl + • 
• + 9n£ 
2 
n 
and to denote the partial differential-quotient of ffx x , x 2 , . . . , x n ) with 
respect to x by f p (x k ), thus giving 
i /*(»*) = a /a9 f£i + + • • • + akn9nL • 
