1881.] particularly those of Two Variables. 209 



(a.) The expressions for the four pairs of conjugate periods, two 

 actual and two quasi ; 



(y3.) The product theorem of Section I is obtained by means of the 

 product theorem proved for single theta-functions by Professor Smith 

 in the paper previously mentioned ; 



(7.) By means of the differential equation which 6 is known to 

 satisfy (see " Cayley's Elliptic Functions," § 310), it is proved that 

 the general function <J> satisfies two equations in x, y of the form 



^-2^-^+2^=0, 

 da>* V KJdx elk 



h t h', E having the usual connexion with M) a. (x). These equations 

 are also investigated from the definition as well as 



d® . 2KA d*<S> A 

 r + =0, 



dr 7T 3 dx dy 



which it is obvious from the theorem of this section that $ satisfies. 



(<5.) Expressions for all the constants occurring in the expansions 

 of all the functions in powers of x, y are obtained. If we write 



^0 = C 0-^j( B ) 5 B 0.l ) ]B ) 2)(^2/) 2 + • ■ • 



+ "^T" (ISro ' ' Nai ' • ' No> ' ' * ' N °- 2 " )( *' y)2 ' l+ 



it is proved that 



c =A 1 . K*A*, 



* > nJ -\K) \a) dp'—dj> 



where jp'=logp, 2'=log#5 r' = 21ogr, 



with a similar expression for A 2 . 



Section III forms the expression of the addition-theorem. Although 

 no addition- theorem proper exists for theta-functions, that is to say, 

 although <I> (x + i;, y + i]) cannot be written down in terms of func- 

 tions of x, y and of £, 77, an expression is obtainable in every case for 



<I>, being either the same or different functions. Since any one 



