y 04 
= 0, 
mall, 
mply 
mear 
inear 
> c 
form 
704] 
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS. 
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 ^ 0 S- 4 out of the upper 
compartment or the lower compartment of any column of the table, the corresponding 
differential combination S- o 0Sr 4 — 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 
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 — b) i {(A + yy) Vcdefa / b / + (A + yx) f c/i^f’ab}, 
where A, fi are constant coefficients: the factor (a — b) has been introduced for con 
venience, as will appear. 
We have consequently a relation 
Vaa / 0Vbb / — Vbb / 0Vaa / = —q- {(A + fiy) Vcdefajb, +(A + fix) Vc / d / e / f / ab}, 
d d 
where, as before, 0 is used to denote u + v‘' ^, u and v being arbitrary multipliers; 
considering u, v as functions of x, y, we have 
d _dx d dy d 
du du dx du dy ’ 
d dx d +dy d 
dv dv dx dv dy ’ 
and thence 0 = P 4- + q4, if for shortness P and Q are written to denote v! ~ +v'~ 
dx dy du dv 
and respectively. 
142. The left-hand side then is 
= p S VBF - - VE1B ' S + Q ( Vaa - Ty - VSF ' I Vaa ') ; 
C. X.