1901 - 2 .] Dr Muir on the Theory of Orthogonants. 
251 
— recourse is had to the two original sets of six equations. In the 
first equation of each set a 2 occurs, in the second /3 2 , and in the 
sixth a/3. Eliminating these in succession we have 
(L - M)a' 2 + (L - N)a" 2 = L - A , 
(L - M)/3' 2 + (L-N)/T 2 = L-B, 
(L — M)a(3' + (L — N)a"/3" = L cos v — c ; 
and thence 
{L-M)(L-N)(a , y8 ,, -a / '/5') 2 = (L - A)(L - B) - (Lcosv-c) 2 ; 
so that one of the nine unknowns 
A)(L - B) - (L cos v - 
(L - M)(L - N) 
^) 2 
the others being like it, and indeed derivable from it, although 
Jacobi does not say so, by cyclical permutation of triads of letters. 
The solution thus reached we may formulate as follows : — 
The Cartesian equation 
Aic 2 + By 2 + Cz 2 + 2ayz + 2bzx + 2cxy = 0, 
where the axes are inclined to one another at angles X, /x, v, may be 
transformed into 
+ + - 0, 
where the axes are rectangular , by means of the substitution 
(L - B)(L - C) - (L cos a. - a) 2 
a 2 (L - M)(L - N) 
h 
/ (M - B)(M - C) - (M cos 
l A 2 (M - N)(M - L) f v 
f (N - B)(N - C) - (X cos a. - a) 2 ’I i 
r a 2 (n-l)(« -m) . j c 
f (L - C)(L - A) - (L cos ju - bf ~\ \ 
\ a 2 (L-.M)(L-N) 
f (L - A)(L - B) - (L cos v - c) 2 ^4 
l a-(L-M)(L-N) 
£ + 
ivhere L, M, hi are the roots of the equation 
x - A 
X COS V 
- c 
x cos fx - b 
X COS V — c 
X 
-B 
x cos X — a 
x cos ix -b 
x cos X 
— a 
x~C 
1 
COS V 
COS fX 
A 2 
t= 
COS V 
1 
cos X 
COS V 
cos X 
1 
