1903-4.] Dr Muir on Orthogonal Transformation of a Quadric. 169 
adduced to show that whether p be a positive or negative integer 
the coefficient of x K x ^ is a rational function of the coefficients of 
the original quadric. 
With this general statement of the case before us, let us take 
up the individual results in order, and see what is obtainable 
therefrom in the light of later work. 
(2) The primary result is the transformation implied in the 
equation 
\XkX\ = bqyp + + . . . + 
kA. 
This, for our purpose, it is essential to write in a form which 
brings into evidence the matrix M of the discriminant of the 
quadric, viz., in the form 
^1 
x 2 
x 3 
= <W + G*,* + 
a n 
a i2 
«13 
Xi 
a 2 i 
a 22 
a 23 
x 2 
a 31 
a 32 
a 33 
x 2 
where, merely for shortness’ sake, oidy three variables are taken. 
Now, as Jacobi himself showed, any equation which holds between 
the a?’s and f s will still hold if we put 
( «n 
a l2 
a i3 . 
)(x L ,x 2 ,x 3 ) for x 1 ,x 2 ,x 3 
a 21 
a 22 
a 23 
a 31 
a 32 
Cl 33 
and Q\V \ , 6 * 2 ^ 2 , G 3 y 3 for y 1 , y 2 , y 3 . 
This substitution, however, in the bipartite function on the left 
results simply in the matrix of the discriminant being twice 
multiplied by itself,* so that we have 
x, x n 
= G x V + <W + g 3 V- 
M 3 
Trans. R. S. Edinb., xxxii. p. 480. 
