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A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
[704 
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83. The upper characters of the ©’s have thus the values 0, 1, f; 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 + i, as follows : 
© 
0 
7 + 2 
1 
© 
©^,9=-© 
7 + 2 7 
©* 9 = t©*, 
7 +- 2 7 
7 + 2 7 
©2 =-¿©2 
7 — 2 7 
®^ + 2 = 
3 
2© 2 , 
7 
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 
3 1 3 1 
© 2 _ I =(-¿.-¿0* 1=) 
7 +• 2 8 +• 2 7 8 
3 i 
(fi) ^ 2 
7 8‘