MATHEMATICS IN THE NINETEENTH CENTURY 479 



differential equations with uniform solutions. The brilliant researches 

 of Poinleve" deserve especial mention. 



Another field of great importance, especially in mathematical 

 physics, relates to the determination of the solution of differential 

 equations with assigned boundary conditions. The literature of this 

 subject is enormous; we may therefore be pardoned if mention is 

 made only of the investigation of our countrymen Bocher, Van 

 Vleck, and Porter. 



Since 1870 the theory of differential equations has been greatly 

 advanced by Lie's theory of groups. Assuming that an equation or a 

 system of equations admits one or more infinitesimal transformations, 

 Lie has shown how they may be employed to simplify the problem 

 of integration. In many cases they give us exact information how 

 to conduct the solution and upon what system of auxiliary equations 

 the solution depends. One of the most striking illustrations of this 

 is the theory of ordinary linear differential equations which Picard 

 and Vessiot have developed, analogous to Galois 's theory for algebraic 

 equations. An interesting result of this theory is a criterion for the 

 solution of such equations by quadratures. As an application, we 

 find that Ricatti's equation cannot be solved by quadratures. The 

 attempts to effect such a solution of this celebrated equation in the 

 century before were therefore necessarily in vain. 



A characteristic feature of Lie's theories is the prominence given 

 to the geometrical aspects of the questions involved. Lie thinks in 

 geometrical images, the analytical formulation comes afterwards. 

 Already Morge had shown how much might be gained in geometrizing 

 the problem of integration. Lie has gone much farther in this direc- 

 tion. Besides employing all the geometrical notions of his predeces- 

 sors extended to n-way space, he has introduced a variety of new 

 conceptions, chief of which are his surface element and contact 

 transformations. 



He has also used with great effect Pliicker's line geometry, and his 

 own sphere geometry in the study of certain types of partial differential 

 equations of the first and second orders which are of great geometrical 

 interest, for example, equations whose characteristic curves are lines 

 of curvature, geodesies, etc. Let us close by remarking that Lie's 

 theories not only afford new and valuable points of view for attack- 

 ing old problems, but also give rise to a host of new ones of great 

 interest and importance. 



Groups 



We turn now to the second dominant idea of the century, the 

 group concept. 



Groups first became objects of study in algebra when Lagrange 

 (1770), Ruffini (1799), and Abel (1826) employed substitution groups 



