66 Obituary Notices of Fellows deceased. 



exhausting, piece of work. The labour lasted for nine years. During 

 that time, Engel's co-operation and assistance were given, without 

 stint and in a loyalty beyond praise, and fully merited the acknowledg- 

 ment which Lie made in his preface. The result was the ' Theorie der 

 Transformationsgruppen,' a treatise in three volumes, covering over 

 two thousand pages, the contribution to science by which his name 

 will probably best be known. It is a work of great originality, con- 

 taining many methods and a wide range of development; it exhibits 

 in masterly manner the suggestive application of new methods to 

 fundamental subjects ; and it may be described briefly as a systematic 

 exposition of Lie's investigations on groups of transformations that are 

 continuous and finite. Among the subjects to which application is 

 made, may be mentioned the theory of ordinary differential equations ; 

 the theory of partial differential equations, both single and in systems ; 

 differential invariants and their types ; the solution of Pfaff's problem ; 

 tangential transformations, especially in spaces of two and three dimen- 

 sions, and more generally in n dimensions ; groups of functions trans- 

 formable into one another, and a substantial simplification (by the use 

 of their properties) in the integration of systems of partial differential 

 equations ; a complete determination of types of the groups of trans- 

 formation in one, two, and three variables, and a partial determination 

 of those in n variables. It concludes with a profound study of the 

 foundations of geometry from the point of view of Biemann and 

 Helmholtz ; and after a critical discussion of the significance of the 

 hypotheses which they made, he propounds a solution of his own, based 

 upon more elementary hypotheses. 



While this work was in progress, Lie changed the scene of his life 

 by accepting, in 1886, the Chair of Mathematics at Leipzig, which had 

 been vacated by Klein on his appointment at Gottingen; Engel 

 accompanying him, and soon being nominated a colleague. Such a 

 professorship possessed some obvious advantages for Lie as compared 

 with the somewhat isolated chair at Christiania. It secured him a 

 wider recognition ; it gave him an audience ; it offered him the chance 

 of able pupils, who would work sympathetically in development of his 

 mathematical theories. Though these advantages did not come early 

 enough to encourage him, still they did come gradually, and some of 

 them in full measure. His work began to be known better and to be 

 appreciated; his methods began to influence mathematicians. Pupils 

 came to him from far and near, and one in particular, George Scheffers, 

 rendered to him offices similar in kind to those rendered by Engel. 

 When once the merit of his work began to be recognised, scientific 

 honours were bestowed upon him freely. He received the honorary 

 or foreign membership of societies and academies in great numbers ; 

 in particular, he was enrolled among our Foreign Members in the 

 year 1895. 



