PARTICULARLY THOSE OF TWO VARIABLES. 
785 
INTRODUCTION. 
Literature of the Subject. 
1. The published investigations on the double theta-functions may be conveniently 
divided into four classes :— 
(i.) Those concerning the algebraical integrals of the equations 
da dy da 
A /X i ~ v /Y" h A /Z 
xdx t ydy , zch _ A 
where X, Y, Z are of the form 
or 
(ii.) Those concerning the theta-functions, properly so-called, proceeding from the 
definitions and investigating the relations which hold between functions of 
different (and of the same) arguments ; 
(iii.) The transformation theory; 
(iv.) The applications to geometry, principally in reference to Rummer’s 16 -nodal 
quartic surface. 
2. The principal papers are :— 
For (i.) I. Abel. His chief memoir is one occurring in the ‘ Memoires des 
r 
Savans Etrangers,’ t. vii., 1841 (but presented to the French 
Academy in 1826), under the title “ Memoire sur une propridtd 
generate d’une classe tres-etendue de fonctions transcendantes,” 
pp. 176 sqq.; the particular case of (i.) is considered p. 260. Several 
other papers, less important, on the transcendental functions occur in 
the collected edition of his works. 
II. Jacobi, (a) “Considerationes generales de transcendentibus Abelianis,” 
‘ Crelle,’ t. ix. (1832), p. 394; 
(y3) “ De functionibus quadrupliciter periodicis quibus theoria 
transcendentium Abelianarum innititur,” ‘Crelle,’ t. xiii. (1835), 
p. 55; 
(y) “ Demonstratio nova theorematis Abeliani,” c Crelle,’ t. xxiv. 
(1842), p. 28. 
III. Bichelot. “ Ueber die Integration eines merkwiirdigen systems 
Differentialgleichungen,” ‘Crelle,’ t. xxiii. (1842), p. 354. 
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