62 Obituary Notices of Fellows deceased. 



From that date onwards, the history of the man is mainly the 

 history of his ideas ; the external incidents of his life are compara- 

 tively few. 



At the beginning of 1871 he was assigned a junior post in his own 

 University, and in the summer of that year he graduated as Doctor. 

 The thesis then submitted was subsequently amplified and became his 

 famous memoir,* "Ueber Complexe, insbesondere Linien- und Kugel- 

 Complexe, mit Anwendung auf die Theorie partieller Differential- 

 Gleichungen. " In this memoir he constructs the theory of tangential 

 transformations! for space ; he applies it to partial differential equa- 

 tions of the first order ; he develops the transformation of Pliicker's 

 line-geometry into a sphere-geometry, which now is regularly associated 

 with his name ; and he shows how the results can be applied to 

 ordinary differential geometry, obtaining (among other properties) the 

 result that his transformation of line-geometry into sphere-geometry 

 makes the asymptotic curves of one surface correspond to the lines of 

 curvature of the ' transformed surface. These are but a few of the 

 results in a paper which is full of powerful methods and novel ideas ; 

 they are sufficient to show that the man who, before 1868, was 'hesitating 

 about his vocation in life, had found an effective vocation by 1871. 



In the succeeding year, the Norwegian Storthing was induced to 

 create a special professorship for him in the University of Christiania. 

 His appointment as Professor Extraordinarius in 1872 enabled him for 

 the future to devote himself to his researches, free from the distracting 

 necessity of supplementing the over-modest salary of his earlier post 

 by private teaching. 



About this time Lie seems to have made his first discovery as to the 

 relations that can subsist between ordinary differential equations and 

 infinitesimal transformations ; the scope of such a relation can be 

 indicated by the simple example of an equation of the first order. A 

 function 12(, y) is said to admit a finite continuous group of transforma- 

 tions represented by 



*'Math. Ann./ vol. 5 (1872), pp. 145256. 



t The transformation of surfaces adopted makes (not merely a point correspond 

 to a point, but) an element of any surface at a point correspond to an element of 

 the transformed surface at the corresponding point. The property holds over the 

 whole of the two surfaces, and, for instance, in the case of ordinary space, leads to 

 the analytical relation 



dz'p'dx' q'dy' = p (dz pdx qdy), 



where x, y, z, p, q, define an element of the one surface, a?', y', z', p', q', define the 

 corresponding element of the transformed surface, and p is a non-vanishing quantity 

 that does not involve differential elements. Such a relation is the basis of the 

 analytical theory of tangential (or contact) transformations. 



