1903-4.] 
Dr Muir on General Determinants. 
79 
(1779), Gauss, Binet, Cauchy (1812), Lebesgue, Jacobi (1841), and 
Cauchy (1841). 
The first section of the paper is said to deal with “ the pro- 
perties of determinants considered as derivational functions .” As 
a matter of fact, however, a close examination shows that the 
functions whose properties are investigated are not strictly deter- 
minants, but belong to a class afterwards named bipartites by 
Cayley himself. It is true that it is the determinant notation 
which is employed in specifying the functions, but this is due to 
the fact that the bipartite under discussion is of a very special 
type, and so happens to be expressible as a determinant. 
The function U from which he considers his three determinants 
to be “ derived ” is 
x(a£ + /3rj + ... . ) 
+ x\a£ 4 - fS'rj + . • . . ) 
+ 
there being n lines and n terms in each line. This at a somewhat 
later date (1855) he would have denoted by 
(a fi 17, a', ... ) 
a ft .... 
and called a bipartite. A still later notation is 
£ v — 
a /3 ... . X 
a ft ... . X ' 
from which each term of the final expansion is very readily 
observations.” it may be well to mention that the substance of the only 
sentence in it which concerns us had already appeared in the memoir of 1773. 
The sentence is 
“ De la il s’ensuit aussi qu’on aura 
{t"m - t'u ") 2 = (jbV - x'z ") 2 + (y"z' - y'z"f + {x"y' - x'y"f, 
= (x ' 2 + y ' 2 + z' 2 ) (x " 2 + y " 2 + z" 2 ) - (x'x" -f y'y" + z'z") 2 . ” 
— JVouv. Mem. de V Acad. Roy ( Berlin ), ann. 1778, p. 160. 
