700 SCIENTIFIC THOUGHT. 



The peculiarity of such dependence, as exemplified in 

 the phenomena of the steady How of heat or of electric 

 distribution, consisted in this, that if at certain points 

 or in certain regions of space the thermal or electrical 

 conditions were defined and known by actual observation, 

 then the whole distribution in other points and regions 

 was completely determined. Those boundary conditions 

 could therefore be regarded as the necessary and sufficient 

 definition of the whole existing distribution. Translated 

 into mathematical language, this means that functions 

 exist which are completely defined by boundary values 

 and singularities i.e., values at single points. Nature 

 herself had shown the way to define and calculate 

 measured relations when through their intricacy they 

 evaded the grasp of the ordinary operations of algebra. 1 

 Pliicker had already in geometry (following in the lines 

 of Newton), when attacking the problem of the infinite 

 variety of higher curves, suggested the method of classi- 

 fying them according to their characteristic properties 

 or singularities. What had been done by geometers 

 and physicists in isolated cases with the expenditure 

 of much ingenuity and skill, Eiemann and his school 

 elevated to the rank of a general method and doctrine. 



functions acquires a great degree ' the theorem called by Clerk-Maxwell 



of clearness and connectedness, "Thomson's theorem" ('Cambridge 



which is mainly gained by concep- and Dublin Mathematical Journal,' 



tions derived from the (physical) ! 1848, or 'Reprint of Papers on 



theory of the potential, and thus i Electro - statics,' &c., p. 139); and 



exhibits the intimate relationship abroad ' Dirichlet's Principle,' after 



Riemann (1857). Further, Brill 



of these theories " (Bacharach, 

 ' Geschichte der Potentialtheorie, ' 

 Gottingen, 1883, p. 71). 



1 On this subject see Burkhardt's 

 ' Memorial Lecture on Riemann ' 



and Nother's "Bericht" ('Math. 

 Ver.,' vol. iii. p. 247) ; and lastly, 

 a very suggestive address by Prof. 

 Klein (" On Riemann's Influence on 



(Gottingen, 1892), p. 5. &c. ; Bach- : Moder n Mathematics") to the meet- 

 arach (loc. cit.}, p. 30, &c. The i ing of the German Association in 

 latter especially with reference to , Vienna in 1804 ('Report,' p. 61). 



