s 
324 PROFESSOR A. R. FORSYTH OH 
Some preliminary considerations (in connexion with §§92 sqq. of Salmon’s ‘Higher 
Algebra’) are adduced in reference to the eliminants of the three equations in each of 
the variables; thus if X be the equation in x obtained by eliminating y and z, it is 
expressed in the form 
X=B m F m -)- B ;i F w + BpFp 
which afterwards proves useful. Then the ordinary case (above referred to) of Abel’s 
theorem is treated on the lines laid down in Clebsch and Gordan’s £ Treatise on the 
Abelian Functions;’ and under the guidance of this the more general form is investi¬ 
gated with the result 
© being the symbol introduced by Boole. 
The remainder of this section is occupied with the discussion of two examples of this 
theorem. In the first, expressions are obtained for E(?q + %+w 3 ) and n(w 1 +w 3 -f w 3 ), 
E and IT being the second and third elliptic integrals ; and in the second example 
E(«i+ + • • • + Urj) is considered. 
In Section II. the addition-theorem for the functions presented in Weierstrass’s 
memoir (‘ Crelle,’ t. lii., (1856), p. 285) is investigated. [It may be pointed out that 
the fundamental equations occur as natural examples of the more general form of 
Abel’s theorem proved in Section I.; but the equations which are obtained almost 
immediately are identical with those used by Weierstrass, and so this case does not 
belong distinctively to the form of Abel’s theorem for the curve of double curvature.] 
For the purpose of the section use is made of the '‘integral-function,” the partial 
differential coefficients of which with respect to the amplitudes give the squares of the 
Abelian functions. The theory is worked out at some length, and the necessary 
formulae are deduced from the fundamental equations in a manner somewhat different 
from that of Weterstrass. From the form first obtained for the sum of three 
integral-functions an important theorem is deduced in § 21, and a verification of this 
is afterwards furnished by the expansion of the two sides of the equation. It is then 
applied, as already mentioned, to obtain the addition-theorem for the functions. 
In §§ 25, 26 is given the discussion of a particular case of the above, viz., when the 
functions are of the order 2, the fifteen functions being the quotients of all but one of 
the double theta-functions by that one. This has already formed the subject of a 
paper by Cayley in ‘ Crelle,’ t. lxxxviii. (1878), p. 74. 
Section I. 
I. Before proceeding to the consideration of the theorem it is necessary to indicate 
the form in which the eliminant of three equations in three variables (or in general 
of fx equations in p variables) will be used. 
