274 Proceedings of Royal Society of Edinburgh. [sf.ss. 
2 2 o 2 ,2 „2 
x y z = s + $ + s 
associated with the substitution 
X = as + a s' + as" ] 
y — fts + ft s + ft s i 
z = ys + ys + y"s" J 
entails the six relations 
a 2 + 
f? 
+ 
9 ! 1 
y i 1 > 
aid' + 
PP' 
+ 
yY 
= o, 
2 
a + 
d 
+ 
/ 1 
a' a + 
P'P 
+ 
YY 
= o, 
" 2 . 
a + 
d 
+ 
y" - = 
aa + 
pp 
+ 
YY 
= 0; 
that from these and the given substitution we obtain the reverse 
substitution 
s H ax + fty + yz , j 
s = a'x + ft'y + yz, 
s' = d'x + /3"y + y'z ; J 
and that this latter substitution when taken along with the original 
condition gives the second set of six relations 
a + a" + a"" = 1 , fty + ft'y' 4 - ft"y" = 0 , 
ft + ft' + ft" — 1 , ya + yd + y"a" = 0 , 
y + y + y"~ ==• 1 , aft + a! ft' + aft" = 0 . 
Further, it is pointed out that if we put 
€ for a(ft'y" - ft"y) + ft(y d' ~ y"a!) + yW ft" ~ a "ft') 
the ordinary solution of the given substitution results in 
es §= x(ft'y" - ft"y) + y( ya - y"d) + z( aft" - a" ft') \ 
eS = x(ft"y - fty" ) + y{y" a - ya" ) + z( a" ft - aft" ) l 
€S " = X (fty - fty ) + y(ya - y'a ) + z( aft' - aft' ) j , 
and that a comparison of this with the reverse substitution as 
already obtained produces 
ea = fty" 
-Fy, 
ea 
| ft"y 
- fty", 
ea" - fty' 
-Py 
eft = ya" 
- y'a-', 
*P 
= y"a 
- ya-", 
eft" = ya' 
-y'a 
"s 
II 
W 
- a." ft', 
£ 7 
= a" ft 
- a ft", 
ey" — aft' 
- aft 
In the next place it is noted that with the help of these the left 
side of the identity 
(y"a - yd')(aft' - a ft) - (ya - y'a) (a" ft - aft") = ae 
