287 
1909-10.] Dr Muir on the Theory of Orthogonants. 
formation” with him is not as with his predecessors a transformation 
which merely changes 
x 2 + y 2 + 4 2 + . . . into £ 2 + rj 2 + £ 2 + . . . , 
but one which at the same time changes 
ax 2 + by 2 + cz 2 + . . . + 2 fyz + 2 gzx + 2 lixy + . . . into A£ 2 + By 2 + C£ 2 + . . . 
In the second place, he has a fresh mode of arriving at the equation for 
determining A, B, C, . . . Calling the four quadrics just mentioned 
V, V', U, U', he forms the discriminant of U — AY, and asserts that the 
coefficient of all the several powers of A in it must he invariants, and that, 
therefore, if the said discriminant be put equal to 0 and the equation so 
obtained be solved for A, the roots resulting must be identical with the 
roots of the equation 
Discrim. (IT - AY) = 0 ; 
in other words, that we must have identically 
a- A 
h g . . . 
A-A 
0 
0 ... 
h 
b- A f ... 
0 
B-A 
0 ... 
9 
/ c-A . . . 
0 
0 
• o 
1 
so that A, B, C, . . . are the values of A in the equation 
Discrim. (U - AY) = 0. 
Hesse, O. (1859, October). 
[Neue Eigenschaften der linearen Substitutionen welche gegebene 
homogene Functionen des zweiten Grades in andere transformiren 
die nur die Quadrate der Variabeln enthalten. Crelle’s Journ ., 
lvii. pp. 175-182: or WerJce , pp. 489-496.] 
Hesse’s object is that of Rummer (1843), Jacobi (1844, March), and 
Borchardt (1845, January), namely, to prove the reality of the roots of 
Lagrange’s determinantal equation by showing that the product of their 
squared differences is essentially positive. 
Taking the linear substitution 
& 
a k\ x i + <** 2^2 + 
; *=i 
we readily see that . . . £ n is expressible as a sum of terms of the 
