AND DOUBLE THETA-FUNCTIONS. 
997 
this is one of a system of 120 equations; viz.: referring to the foregoing table of the 
120 pairs, it in fact appears that taking any pair such as out of the upper com¬ 
partment or the lower compartment of any column of the table, the corresponding 
differential combination d 0 hd 4 — 3-f>3- 0 is a linear function of any two of the four pairs 
in the other compartment of the same column. 
Differential relation of u, v and x, y. 
141. We have as before, in the two notations, the pairs 
A . B 11 . 7 
C . DE 5.9 
D . CE 13 . 1 
E . CD 14 . 2 
F . AB 10 . 6 
and from the expressions given above for the four pairs below the line, it is clear that 
any linear function of these four pairs may be represented by 
(a-- 6 )^{(X+/ry) \/ cdefa / b / +(X+/xx) \Zcffef ab}, 
where X, /x are constant coefficients, and the factor («— b) has been introduced for 
convenience, as will appear. 
We have consequently a relation 
\/ aa^v/bb, — ffbbbff aa / =^i — {(X+p-y) v / cdefa / b / + (X+p.,r:) \/ c/l^fab}, 
where as before 6 is used to denote u and v being arbitrary multipliers ; 
considering u, v as functions of x, y , we have 
<f 
die 
(f 
dv 
dx d dy d 
du dx' die dy 
dx d dy d 
—L j 
dv dx dv dy 
5 
and thence 5 = P 7 - + Q ~ if for shortness P, Q are written to denote u'—A-v'^r and 
dx dy du dv 
respectively. 
