423 
1907-8.] Dr Muir on the Theory of Hessians. 
are in constant use by him. It is clear, however, that the determinant 
which he had in mind was not Hesse’s, but that which the year following 
he named the “ discriminant.” * 
The interest of the present paper lies in the fact that, amid much other 
matter, not only are the said two determinants clearly defined and dis- 
tinguished, but are shown to be viewable as having a common parentage, 
being indeed two extreme members of a family group. In the first place, 
the determinant of any homogeneous integral function is incidentally defined 
as the resultant of the first partial differential coefficients of the function, 
when drawing attention to Boole’s proposition (1843) that the said 
determinant “ is unaltered by any linear transformation of the variables, 
except so far as regards the introduction of a power of the modulus of 
transformation.” It is spoken of later in the paper as the “common 
constant determinant” or the “ordinary determinant” of the function, 
the word discriminant not being proposed until a later date in the same 
year. In the second place, there is brought into notice in connection 
with any homogeneous integral function <f>(x , y , . . . , z) of the n th degree 
the family of functions 
( 4 + 4 + 
where r has the values 1,2,. . . , n. Corresponding to these there is a 
family of determinants (i.e. discriminants), namely 
where r— 2,3,. . . , n , the first being according to Sylvester the “ Hessian ” 
or “ First Boolian ” determinant f of f>, and the last the “ Final Boolian ” or 
“ordinary determinant” of <p. The reader is left in the former case to 
reconcile the new definition with Hesse’s own definition, and in the latter 
case to observe that 
* See Philos. Magazine , ii. (1851) p. 406, and Cambridge and Dub. Math. Journ ., vii. 
(1852) p. 52 ; or Sylvester’s Collected Math. Papers, i. pp. 280, 284. 
t On p. 194 he says the Hessian of F(a;, y ) is “the determinant of the determinant , in 
respect to | and 77, of 
— an error which is repeated in the Collected Math. Papers. 
