2 77 
1909-10.] Dr Muir on the Theory of Orthogonants. 
the r th column of the adjugate of A as multipliers in connection with the 
given set of equations and performing addition, when there results the r th 
equation of the required set.* 
Hesse, O. (1851, April). 
[Ueber die Eigenschaften der linearen Substitutionen, durch welche 
eine homogene ganze Function zweiten Grades, welche nur die 
Quadrate von vier Yariabeln enthalt, in eine Function von 
derselben Form transformirt wird. Crelles Journ., xlv. pp. 93- 
101 : or Werke, pp. 307-317.] 
Starting with the supposition that the substitution 
makes 
y k = a kl X x + a k2 X 2 + . . . + a kn x n 
b x y\ + b 2 y\ + . . . + b n yl = a x x\ + a 2 x\ + . . . + a n x\ , 
Hesse obtains by differentiation with respect to x 1 , x 2 , x 3 , x 4 the reverse 
substitution 
j Tc—n 
^k^k = “t "t • • • F Q-nkbnV n ( ) 
) k= 1 
and having thus found that the latter substitution will make 
a x x\ + a 2 x 2 2 + . . . + a n x l n = hy\ + b 2 yl + . . . +b n y 2 n 
he is able by putting rj k for a k x k and £ k for b k y k 
to say that the substitution 
| k—n 
Vk = <*-lk€l + a 2k%2+ • • • +^nk^k f 
; l 
* It should be noted that the theorem holds when A is any determinant whatever. 
Further, there is implied in it another of at least equal importance, namely : — If a stand for 
| a 11 a 22 . . . a nn | , the equation whose roots are A times the reciprocals of the roots of the equation 
a xl -9 
a 12 • • 
ain 
a 2l 
a 22 - 9 . . 
a<in 
a n i 
a n 2 . • 
a nn - 0 
A u -® 
Aj 2 • • 
• Ai n 
A 2 i 
A 22 -0 . . 
A 2 n 
= 0 
A«i 
A ri 2 
• • A nn — © 
An independent proof of this is readily obtained by substituting a/© for 9 in the original 
equation, expanding the determinant in a series arranged according to descending powers 
of a/0 , using © m /a as a multiplier, substituting A n , A 12 ... for their equivalents, and 
returning to the determinant form. 
