692 



SCIENTIFIC THOUGHT. 



conception of the variable ^ has nndergone in the course 

 of the last hundred years. Here we come upon a 

 term which was introduced into mathematical language 

 mainly through the writings of Euler — the term 

 function. It is used to denote the mathematical 

 dependence of two or more variable quantities on each 



1 To the theory of equations in 

 algebra there corresponds the 

 theory of differential equations in 

 analysis ; and as the theory of 

 algebraical equations had gradually 

 emerged in a complete form out of 

 investigations of special equations, 

 or sets of equations, so likewise in 

 analysis a general theory of differ- 

 ential equations is gradually being 

 evolved out of the scattere'd and 

 very extensive investigations of 

 special differential equations which 

 presented themselves notably in 

 the apijlication of analysis to astro- 

 nomical and physical problems. It 

 is claimed by those who have 

 grasped the abstract ideas of 

 Sophus Lie, that he has taken a 

 great step forward in the direction 

 of a general theory of differential 

 equations, by apjjlying methods 

 which suggested themselves to him 

 through the general theory of alge- 

 braic forms and its C(jnnection with 

 geometry. Accordingly, the the- 

 ories of Lie can be termed an 

 algebraical theory of differential 

 equations, depending upon trans- 

 formations analogous to those 

 which had been established in the 

 general theory of forms or quantities 

 of which I treated above. Prof. 

 Engel, in his obituarj' notice of 

 Sophus Lie ('Deutsche Math. Ver.,' 

 vol. viii. p. 35), tells us that in the 

 year 1869-70, when Lie met Prof. 

 Klein in Berlin, the former was 

 occupied with certain partial differ- 

 ential equations which exhibited, 

 under certain transformations, in- 

 variantive properties, and that Klein 



then pointed out " that his pro- 

 cedure had a certain analogy with 

 the methods of Abel. The sug- 

 gestion of this analogy became im- 

 portant for Lie, as he was generally 

 intent upon following up more 

 clo.^ely the analogies with the 

 theory of algebraical equations." 

 Dr H. F. Bakei-, in his recent article 

 on Differential Equations in the 

 'Ency. Brit.' (vol. xxvii. p. 448), 

 roughly distinguishes two methods 

 of studying differential equations, 

 which he names respectively 

 " transformation theories " and 

 "function theories," "the former 

 concerned to reduce the algebraical 

 relation to the fewest and simjjlest 

 forms, eventually with the hope of 

 obtaining explicit expressions of 

 the dependent in terms of the 

 independent variables ; the latter 

 concerned to determine what gen- 

 eral descriptive relations among 

 the quantities are involved by the 

 differential equations, with as little 

 use of algebraical calculations as 

 may be possible." For the historj' 

 of thought and connection of ideas, 

 it is interesting to learn, through 

 Prof. Engel, that it was not purely 

 algebraical work, — such as is rep- 

 resented by Galois and Jordan, 

 to which Lie was early intro- 

 duced by Pi'of. Sylow, — but the 

 study of Poncelet's and Pliicker's 

 methods which led Lie to his 

 original conceptions, and that he 

 was fond of calling himself a pupil 

 of Pliicker, whom he had never 

 seen (Engel, loc. cit., p 34). 



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