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153. 
A MEMOIR ON THE AUTOMORPHIC LINEAR TRANSFORMATION 
OF A BIPARTITE QUADRIC FUNCTION. 
[From the Philosophical Transactions of the Royal Society of London, vol. cxlviii. for the 
year 1858, pp. 39—46. Received December 10, 1857,—Read January 14, 1858.] 
The question of the automorphic linear transformation of the function x 2 + y 2 + z 2 , 
that is the transformation by linear substitutions, of this function into a function 
xf + y 2 + z 2 of the same form, is in effect solved by some formulas of Euler’s for the 
transformation of coordinates, and it was by these formulae that I was led to the 
solution in the case of the sum of n squares, given in my paper “ Sur quelques pro 
priétés des déterminants gauches”^). A solution grounded upon an à priori investiga 
tion and for the case of any quadric function of n variables, was first obtained by 
M. Hermite in the memoir “ Remarques sur une Mémoire de M. Cayley relatif aux 
déterminants gauches”( 2 ). This solution is in my Memoir “ Sur la transformation d’une 
function quadratique en elle-même par des substitutions linéaires ”( 3 ), presented under a 
somewhat different form involving the notation of matrices. I have since found that 
there is a like transformation of a bipartite quadric function, that is a lineo-linear 
function of two distinct sets, each of the same number of variables, and the develop 
ment of the transformation is the subject of the present memoir. 
1. For convenience, the number of variables is in the analytical formulas taken 
to be 3, but it will be at once obvious that the formulae apply to any number of 
variables whatever. Consider the bipartite quadric 
(a , b , c $>, y, s#x, y, z), 
a', V, c' 
a", b", c' 
1 Grelle, t. xxxii. (1846) pp. 119—123, [52]. 
2 Cambridge and Dublin Mathematical Journal, t. ix. (1854) pp. 63- 
3 Crelle, t. l. (1855) pp. 288—299, [136]. 
-67. 
C. II. 
63