Soplius Lie. 65 



At any rate, whatever the explanation may be, Lie sought relief in 

 change of subject, and devoted himself, almost entirely for the next 

 few years and partially for the rest of his life, to differential geometry. 

 In a long succesion of valuable papers, he made masterly additions to 

 our knowledge of minimal surfaces, particularly those which are 

 algebraic ; he dealt with surfaces which have their Gaussian measure of 

 curvature equal to a constant, or are determined by other assigned 

 relations between their principal radii of curvature ; and he discussed 

 surfaces as generated by the translational motion of a curve. The 

 theory of his groups was frequently applied in these researches, and 

 with considerable effect ; thus his papers on the classification of sur- 

 faces according to the groups of transformations of their geodesies are of 

 high importance. Darboux, in the ' Notice sur M. Sophus Lie,' already 

 quoted, indicated his sense of the value of these contributions to 

 differential geometry : no less significant is the testimony in Darboux's 

 great treatise, ' Theorie generale des surfaces,' furnished by the number 

 of references to Lie's name in its index. 



Yet during this specially geometrical period, he did not altogether 

 neglect the development of his theory of continuous groups ; occasional 

 papers were written from time to time, showing that it still occupied 

 part of his constructive thought. Towards the end of the period, 

 about 1882, his papers gave signs of his having again reverted to 

 differential equations by applying his groups to the classification and 

 integration of ordinary differential equations of any order. Moreover, 

 the publication of Halphen's thesis on differential invariants led Lie to 

 point out that his own earlier work included Halphen's investigations. 

 His attention was thus again turned to the subject, and one conse- 

 quence was that he gave the general theory of differential invariants, 

 not merely for the projective group, as in Halphen's work, and in the 

 subsequent detailed work of a number of English mathematicians, but 

 for any finite continuous group of transformations.* 



Lie's investigations had now extended over a considerable number 

 of years. They had covered a wide range in a variety of subjects, 

 and the results had been published in no consecutive form and in 

 partly inaccessible places. He had from time to time thought of 

 undertaking some treatises dealing with the main topics which had 

 occupied his thoughts for more than fourteen years. But it was not 

 until September, 1884, that any such project took a practical shape. 

 In that month Friedrich Engel came to Christiania, partly in order to 

 make himself acquainted with Lie's work, partly (on the advice of 

 Klein and A. Mayer) to assist Lie, if that were possible, in making a 

 systematic exposition of the whole theory of transformation-groups. 

 It was exceedingly fortunate for Lie that he thus found some active 

 co-operation and steady assistance in the execution of a severe, even 



* 'Math. Ann.,' vol. 24 (1884).. pp. 537578. 



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