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,