174 FURTHER INVESTIGATIONS ON THE DOUBLE ^-FUNCTIONS. [663 
But I have also in the table inserted the values to which — $ 2 2 , — 8^, etc., are 
respectively proportional, viz. the table runs — 8,/= a, — 8^ = b, etc., (read — $ 2 2 = {a], 
— 8* = [b], etc., the brackets { } having been for greater brevity omitted throughout the 
table), and where it is of course to be understood that — S 2 -, — S^, etc., are proportional 
only, not absolutely equal to {a}, {6}, etc. And I have also at the foot of the several 
columns inserted suffixes oo oo, ab, cd, etc., which refer to the columns of Table II. 
Comparing the first with any other column of the table, for instance with the 
second column, the two columns respectively signify that 
— 8o 2 (u) = {a}, 
— 8£ (u + A) — — {be}, 
-8Hu)={b\, 
— Si* {u + A) — - {ae}, 
.e. 
II 
•• ••• 
4 2 (u +A) = - {e}, 
where, as before, the sign = means only that the terms are proportional; u is written 
for shortness instead of (u, u), and so u + A for (u + A, u + A'), etc.: the variables in 
the functions [a], {be}, etc. are in each case x, x. But if in the second column we write 
u~A for A, then the variables x, oc will be changed into new variables y, y', or the 
meaning will be 
X, X 
y> V 
— 8,/ (u) = {a], 
- 8i 2 (u) = - {be}, 
- &! 2 (U) = {6), 
- Si 2 (u) = - {ae}, 
Qi (u) = {ab}, 
.. £> •• 
T 
1 
5C 
so that, omitting from the table the terms ^ 
except only the outside left-hand column — 
vhich contain the capital letters P, Q, R, 8, 
8*, — 8j 2 , etc., the table indicates that these 
functions — $ 2 2 , — $i 2 , etc., are proportional to the functions {a}, {6}, etc., of x, x given in 
the first column; also to the functions — [be], — {ae}, etc., of y, y' given in the second 
column; also to the functions — {ae}, — [be], etc., of z, z given in the third column; and 
so on, with a different pair of variables in each of the 16 columns. 
Thus comparing any two columns, for instance the first and second, it appears that 
we can have simultaneously 
X, x' y, y 
[a] = - {be}, 
{6} = _ [ae], 
{ab} = -{e}, 
(fifteen equations, since the meaning is that the terms are only proportional, not absolutely 
equal), equivalent to two equations serving to determine x and x' in terms of y and y,