xxxvm 



the developable surface that is generated by the osculating plane of a 

 given tortuous curve, and, therefore, also of any developable surface. 

 He greatly extended Salmon's theory of reciprocal surfaces ; and, re- 

 suming a subject already discussed by Schlafli, he produced* in 18(59 

 his ' Memoir on Cubic Surfaces,' in which he dealt with their com- 

 plete classification. Many of his memoirs are devoted to the theory 

 of skew ruled surfaces, or scrolls as he called them. Our knowledge 

 of geodesies, of orthogonal systems of surfaces, of the centro-surface 

 of an ellipsoid, of the wave-surface, of the 16-nodal quartic surface, 

 not to mention more, is due in part to the extensions he achieved. 

 It is difficult to indicate parts of the general theory of surfaces and 

 of twisted curves that do not owe at least something and frequently 

 much to his labours ; a mere reference to the index of a book like 

 Salmon's ' Solid Geometry ' will show how vast has been his 

 influence. 



One group of subjects interested him throughout his life, the 

 theory of periodic functions, in particular, of elliptic functions : it was 

 to the latter that his only book was devoted. But in a subject, the 

 main lines of which were established so definitely before he began to 

 write, f it is impossible, without entering into great detail, to mark out 

 the contributions that are directly dne to him. When a theory is in 

 such a stage as was that of elliptic functions about 1842, the work of 

 one writer sometimes helps to fill the gaps left by that of another, 

 sometimes develops another writer's results from a different point of 

 view ; the composite theory depends, in part, upon the coordination 

 of complementary results. 



Abel's famous paper,! ' Memoire sur une propriete generale d'une 

 classe tres-etendue de fonctions transcendantes,' presented to the 

 French Academy of Sciences in 1826, and unfortunately delayed 

 in publication for nearly fifteen years, attracted Cayley's atten- 

 tion quite early in his scientific career. In 1845 Cayley published 

 his 'Memoire sur les fonctions doublement periodiques,'|| in which 

 he considered Abel's doubly- infinite products of the form 



(x\ 

 1 + - 

 w) 



* ' C. M. P.,' vol. 6, p. 412; ( Phil. Trans.' (1869), pp. 231326. 



t The history will be .found in Casorati, ' Teorica delle f unzioni di variabili 

 complesse,' 1868, and in Enneper, ' Elliptische Functionen, Theorie und Geschichte,' 

 second edition, 1890, where other references are given. 



J ' CEuvres completes d'Abel ' (Christiania, 1881), vol. 1, pp. 145 211. 



The circumstances are recited in 9 of the appendix to the volume, by 

 Bjerknes, ' Niels Henrik Abel, Tableau de sa vie et de son action scientifique ' 

 (G-authier-Villars, Paris, 1885). 



|| ' C. M. P.,' vol. 1, No. 25; 'Lionville,' t. 10 (1845), pp. 385420. 



