(34 Obituary Notices of Fellows deceased. 



the required equation is 



where F is any functional form. Manifestly, such an idea is capable of 

 wide application : under Lie's direction, it proved fruitful in succeeding 

 years. 



Similarly, the integration of partial differential equations of the first 

 order was discovered by Lie to be bound up with infinitesimal tangential 

 transformations under which they are invariantive. This discovery led 

 him to resume the whole problem of the integration of such equations ; 

 and, as the outcome of his investigations, specially built upon the com- 

 pleted analytical theory of tangential transformations, he made two 

 notable advances. One of these consisted in a great simplification of 

 the known method of Jacobi, by affecting a material reduction in the 

 number of quadrature processes ; the other led him to a newvmethod 

 for the solution of Pfaff's problem, which, besides being simpler and 

 shorter than preceding methods, indicated the real functional signifi- 

 cance of the necessary analysis. 



These results, obtained by connecting infinitesimal transformations 

 with widely verging questions in differential equations, prepared the 

 way for the consideration of a problem certain to possess an extensive 

 range, viz., the theory of finite continuous groups of transformations, 

 in general, and without special regard to any particular application. 

 Lie began this work in 1873, and, for the next three years, concentrated 

 upon it all the intensity of his creative enthusiasm : he once spoke of 

 himself as having, during that period, lived only among his groups of 

 transformations. The result was to constitute this theory an inde- 

 pendent subject : begun, as already indicated, from its association with 

 differential equations, and finding in its progress some of its most direct 

 applications in that region; but, as the theory grew, it obtained * 

 wider significance, and the geometrical bent of much of Lie's thought 

 gave it applications within the region of geometry. 



Towards the close of 1877 Lie had completed one stage of these 

 investigations. His conclusions were embodied in a number of memoirs ; 

 many of them were published in a new journal in Christiania, edited by 

 Sars, Miiller, and himself, some of them in the ' Mathematische Annalen,' 

 most of the latter being revised and extended accounts of earlier 

 papers. Apparently, Lie suffered from severe disappointment at the 

 lack of interest so far shown in his work by mathematicians ; his story 

 at this time reads like the occasional experience of the investigator 

 who lives, remote from fellow-workers and unstimulated by eager 

 pupils, voyaging through his sea of thought alone, at the end finding 

 himself weary, isolated, unacknowledged, perhaps therefore dis- 

 couraged, and certainly left uncheered by any confident satisfaction 

 that others are following- him. 



