Concerning Differential Invariants. 



David A. Rothrock. 



During the last forty years ^^olKlerf^l progress has been made in many 

 fields of higher mathematics. One distinet line of investigation has had to 

 do with a iiiicro-scopic examination of the fundamental axioms of the ele- 

 mentary mathematics, of conditions of convergence, of the sufficient condi- 

 tions in the calculus of variations, and so on. Another essential advance 

 has been made by unifying many sejiarate and apparently distinct fields of 

 mathematics under one common law. Among many advances in this latter 

 line of work, none are more important than the work of Sophus Lie. a Nor- 

 wegian, who lived from 1842 to 1890. 



Lie received his doctorate from the T'niversitj' of Christiania in 1865, 

 caring no more for matheiuatical work than for literary or philological 

 work. In fact, he had thought of becoming an engineer ; but receiving an 

 ai)pointmont to a docentship in the university, he turned his attention to 

 the study of advanced mathematics. The real mathematical genius of 

 Lie was aroused by a course of lectures on substitutions by Professor S.v- 

 low. Lie's creative period seems to have extended from 1SG8 to about 1874, 

 during which time he came into possession of the essential features of his 

 epoch-making Theory of Continuous Groups. The remainder of his life 

 was devoted to the elaboration of his early conceptions, and to the appli- 

 cations of his theories. A general development of the higher number sys- 

 tems, a classilication oi' ordinaiy and partial differential etiuations. with 

 methods of their solutions, invariants and covariants, many problems of 

 physics and astronomy, are all treated froHi the standpoint of the con- 

 tinuous group. Below is sketched a brief outline of the continuous group 

 theory of Lie, as applied to di^Terential invariants, and the calculation of an 

 important differential invariant is indicated. 



1. Point Tram^formation. Let x, y be the Oartesiau coordinates of any 

 point iu the plane, and let xi, yi be any point other than x, y. Then 



xi = * (X, y), yi = * (X, y) 



