278 Proceedings of Royal Society of Edinburgh. [sess, 
B'C" - B"C' = (G /3a + G'p'b' + G "p"c)(Gya" + G'yb" + G "y"c") 
- (G ^a" + G'p'b" + G"P"c")(Gya + G'y'b’ + G"y"c) , 
=-■ G'G'X/3'y" - p"y')(b'c" - b"c r ) 
+ G"G (p"y - py" )( c 'a - c"a) 
+ GG' (Py -P'y )(ab"-a"b') 
— G G att 4- G^Ga b + GG'a'c . 
With this we have to compare 
A A , A „ 
aa + o + ^j,a c , 
the result being that we obtain 
GGG" = A , 
and thus reach the first resting-stage on our journey. 
At the outset of the next stage it is found desirable, for brevity’s 
sake, to introduce six additional letters to denote certain functions 
of the known quantities A , A', A", .... viz. 
p for a 2 + B 2 + c 2 , q for A' A" + B'B" + C'C", 
p for A'^ + B /2 + C' 2 , q or A"A + B"B + C"C , 
p" for A" 2 + B" 2 + C" 2 , q" for A A' + BB' + C C\ 
These are said to entail the six identities 
p'p" - q = (B'C" - B"0') 2 + (C'A" - C"A') 2 + (A'B" - A"B') 2 , 
VV - S' 2 = (B"C -BC") 2 + (C"A -CA") 2 + (A"B -AB") 2 , 
VP - ?" 2 = (BC' - B'C) 2 + (CA' - C'A) 2 + (AB' - A'B f, 
qq"~pq = (B"C - BC" )(BC' - B'C ) + (C"A - CA" )(CA’ -C'A ) 
+ (A"B - AB" )(AB' -A'B ), 
q"q -p'q' = (BC' -B'C )(B'C" - B'C") + (CA' - C'A )(C'A" - C"A') 
+ (AB' -A'B )(A'B" - A"B'), 
qq' -p"q" = (B'C" - B"C')(B"C - BC" ) + (C'A" - C"A')(C"A - CA" ) 
+ (A'B" - A"B')(A"B - AB"), 
and 
A 2 = 19V V -M 2 ~PP 2 - P V 2 + ^qqq • 
The original set of nine equations, giving A, A', A", ... in terms 
of the three G’s and the coefficients of the substitutions, is then 
returned to, and the following equations derived, — 
