946 
PROFESSOR A. CAYLEY ON THE SINGLE 
83. The upper characters of the ©’s have thus tire values 0, 1, -§; the lower 
characters are originally 2, 1, 0, or —1, and these have when necessary to be by the 
addition or subtraction of 2 reduced to 0 or 1 ; the effect of this change is either to 
leave the © unaltered, or to multiply it by —1 or Xi, as follows 
^ o 
1+ 
lO 
II 
0 
© 
y 5 
«4+2= 
© 
1 
to 
II 
1 
l 
<S> . 
® 
1 
7 + 2 
i 
-v 
4+2= 
-i®y, 
©7 + 2 = 
3 
i©' , 
y 
where only the first column of characters is shown, but the same rule applies to the 
second column ; and where we must of course combine the multipliers corresponding to 
the first and second columns respectively : for instance 
©7 + 2 l + 2=( — i- 
Thus taking the tenth line of the upper half, and the fifth line of the lower half, we 
have 
10 
01 
1113 . 
2 2 2 2 
3 13 3 
2 2 2 2 
13 11 
2 2 2 2 
3 3 3 ] 
2 2 2 2 
00 
10 
1 0-1 0 
1 0-1 0 
1 0-1 0 
1 0 — 1 0 
giving the value of <9-^ ^{u—u '): viz. this is 
4 
H 2w)-®_ 
2 
-1 
O ti|c 
To 
T 
f© 2 
0-4 
o< 2 “') 
+4 
J(» 
) • © 
3. 
-1 
x>) 
3_ 
— 
1 
2 f 
O' ” 
>•4 
0< ” 1 
1 
3. 
1 
1 
1 
3. 
1 
x 
+ ©J 
q( ” 
).©_ 
2 
-1 
;<•*> 
+i®{ 
o(” 
).©J 
q( » ) 
.3 
o 
3 
i 
3. 
3 
1 
x 
+4 
o(” 
) . © 
2 
-1 
;<»> 
—i© 2 
2 ( 
0 v» 
)•©; 
0 (» ) 
where the first column is the value given directly by the diagram, and which is then 
reduced to that given by the second column. 
84. But instead of the ®’s we introduce single letters (X, Y, Z, W), (E, F, G, H), 
(I, J, K, L), (M, N, P, Q), with the suffixes (0, 1, 2, 3), in all 64 symbols, thus 
