[710 
10] 
ON A DIFFERENTIAL EQUATION. 
25 
The six functions of the set (1), that is, 
1 1 
X, 1 — X, 
X x—1 
1 — X ’ X — 1 ’ X 
form a group; and by operating with the substitutions of this group, and of the like 
group 
1 s z— 1 
1 — z ’ z — 1 ’ z ’ 
X + 1\ 2 
z, 1 —z, 
1 
upon any value of s in the sets (2), (3), (4), for instance upon s=(^-—jJ , we form all 
the 18 functions of these sets. 
In any one of these sets (2), (3), and (4), comparing two forms (the same or 
different), for instance in the set (2), writing y for 2 and then in one form 0 for x, 
or 
s'= («--!)’ and 2' = (i3l) !> whence 
= and ,’/ = <Z -\j ~ > whence 
(X + ly fz + T\ 3 
W-1/ ~\z-l) ’ 
fx+ ly _ (s+ l) 2 
U-l/ ~~ 4s ’ 
we obtain either the equations of the set (1) or those of the sets (8), (9) and (10); and 
whether we use the set (2), (3) or (4), the only new equations obtained are thus the 9 
equations of the sets (8), (9) and (10). These several equations present themselves 
however in different forms: for instance, instead of the equation 
we may obtain 
0-1)= 
4x 
4£ 
(x-1) 2 ’ 
0+1 y 
/ x + ly 
4>z 
U-iy 
If, to get rid of this variety of form, we multiply out the denominators, the 9 
equations are 
0 = x"z 2 — 
2 x 2 z 
— 2xz 2 + 
x 2 — 12 xz + 
z 2 — 2x — 2 z + 
1, 
0 = x 2 z 2 
— 16xz 
+ 16x + 16s - 
16, 
0 = 16x 2 z 2 — 
16x 2 z 
— 16xz 2 
+ 16xz 
- 
1, 
0 = x 2 z 2 — 
2 x 2 z 
+ 
X 2 + 16xz 
- 16s 
y 
0 = 
16x 2 z 
—16xz— 
s - 4- 2s — 
1, 
0 = 
16x 2 z 
- 
16« 2 — 16xz + 
s 2 + 16« 
, 
0 = x 2 z 2 
— 2 xz 2 
-|- lbxz 
s 2 — 16x 
y 
0 = 
16xz 2 — 
X 2 — I6xz 
-p 2x -p 
1, 
0 = 
16xz 2 + 
x 2 — 16xz — 
16s 2 + 16's 
These 9 equations are derivable all from any one of them by the changes of the set (1) 
upon x and z. 
Cambridge, 3rd June, 1879. 
C. XI. 
4