1901 - 2 .] Dr Muir on the Theory of Orthogonants. 
275 
becomes first 
and then 
and that consequently 
•W-jSV) 
e 2 • ea \ 
e 2 = 1 . 
Lastly, attention is very pointedly drawn to the fact that if the 
nine quantities a, a , a, ft , ft', ft", y, y, y" be such as the foregoing 
results imply, and any three quantities X, Y, Z be connected with 
other three P, Q, R by the equations 
X = aP + a'Q + a"R ] 
Y = ftF + ft'Q + ft"R> 
Z = yP + y Q + y"R j 
then it follows that 
P = aX + ftY + yZ ] 
Q = a'X + ft'Y + y'Z [ 
R = a"X + ft'’ Y + y 'Z j 
and 
X 2 + Y 2 + Z 2 = P 2 + Q 2 + R 2 .* (0) 
The next preliminary step is to formulate the equations 
which result from the identity of (Ax + By + C z)to + • • • • with 
G st + G's'^ + G"s 'v . These are f 
A :?= Gatt + G'ab +G "a'c B = G/3a +G ' g!b +G"j3 ”c C = G yCL +G 'y'b +G "y"c 
A' = GaClt + G' o!b' -\-G''a"c' B ? — G^a' -r G 7 /3 V + G" fi"c! C' = Gya' 4- G'y'b' -\-G"y"c' 
A' — Gaa" + G'ab" + (Y a'c" B w = Gj8a" + G f&'b" G” f& "c" G" — Gycc' + G'y'b" + G"y"c' . 
Along with the twelve relations previously obtained, they give in 
all twenty-one equations for the determination of the three G’s, 
and the eighteen coefficients of the substitutions. 
The actual process of solution consists in a long series of 
deductions from the last-obtained set of nine equations, the 
repeated use of the twelve other equations being disguised by 
* In leaving these preliminary deductions, it may be worth remarking 
that the like results which flow from the second given substitution and its 
associated condition are not taken entirely for granted by Jacobi, but are 
given with equal fulness, the two series indeed appearing in parallel 
columns. t v. next page. 
