1903 - 4 .] Dr Muir on Orthogonal Transformation of a Quadric. 171 
With this before us let us return to the primary result 
*1 
x 2 
x 3 
II 
«11 
^12 
U 13 
X l 
a 21 
a -22 
a 23 
X 2 
a 31 
C( 32 
a 33 
X 3 
&iVi + G 2 t / 2 2 + G 3 !/ 3 2 , 
and make use of the theorem * that any equation which holds 
between the x’s and if s will still hold if we put 
( A u 
A 21 
A31 
) (aq , x 2 , x 3 ) for x 1 , x 2 , x 3 
A 
A 
A 
a 12 
A 22 
Ag 2 
A 
A 
A 
A13 
A 2 3 
A33 
A 
A 
A 
and 
Vx 
Gi 
V 2 
*<V 
rr for y-nVvVz- 
(jTg 
The performance 
of the substitution on the left-hand side 
changes the matrix M into M 1 M M 1 , that is, M -1 , and we have 
Xj X , 
2 X 3 
_ V\ , V 2 , l Ja 
Gi G 2 G 3 ' 
x x 
M 
-l 
x 2 
x 3 
The repetition of the substitution upon this equation, and the 
application of the same process to the equation 
X J ^ g 3 . = y* + y* + 0. 
1 . . aq 
1 . x 2 
. . 1 x 3 
* Jacobi’s enunciation of this is “In relationibus omnibus, quae inter 
variabiles aq , x 2 , . . . , x n et variabiles yf y y 2 , . . . , y n locum habent, 
simul loco y m poni posse ff-, atque loco x . 
G m 
~t~ &2A.^2 ~b . . . 4~ bnXpCn ^lA^l 4" ^2A.*^2 4“ . . • + t) n 0JCn jj . 
G 1 G 2 . • . Gr?i, 2 db . Ct>nn 
where the &’s correspond to the modern A’s, and the sign of equality is used 
for £ or. ’ 
