696 



SCIENTIFIC THOUGHT. 



47. 



Physical 

 analogies. 



direction of promoting these seemingly abstract re- 

 searches. Nature herself exhibits to us measurable 

 and observable quantities in definite mathematical de- 

 pendence ; * the conception of a function is suggested by 

 all the processes of nature where we observe natural 

 phenomena varying according to distance or to time. 



1 Nearly all the " known " func- 

 tions have presented themselves in 

 the attempt to solve geometrical, 

 mechanical, or physical problems, 

 such as finding the length of the 

 arc of the ellipse (elliptic func- 

 tions) ; or answering questions in 

 the theory of attraction (the poten- 

 tial function and other functions, 

 such as the functions of Legendre, 

 Laplace, and Bessel, all comprised 

 under the general term of " har- 

 monic functions"). These func- 

 tions, being of special import- 

 ance in mathematical physics, were 

 treated independently before a 

 general theory of functions was 

 thought of. Many important pro- 

 perties were established, and 

 methods for the numerical evalu- 

 ation were devised. In the course 

 of these researches other functions 

 occurred, such as Euler's "Gam- 

 ma " function and Jacobi's " Theta " 

 function, which possessed interest- 

 ing analytical properties. These 

 functions, suggested directly or 

 indirectly by applications of analy- 

 sis, did not always present them- 

 selves in a form which indicated 

 definite analytical processes, such 

 as processes of integration or the 

 summation of series. Very fre- 

 quently they presented themselves, 

 not in an "explicit" but in an 

 "implicit" form; their properties 

 being expressed by certain condi- 

 tions which they had to fulfil. It 

 then remained a question whether 

 a definite symbol, indicating a 

 set of analytical operations, could 

 be found. This arises from the 



fact that the solution of most prob- 

 lems in mechanics and physics 

 starts from the assumption that, 

 though the finite observable pheno- 

 mena of nature are extremely in- 

 tricate, they are, nevertheless, 

 compounded out of comparatively 

 simple elementary processes, which 

 take place between the discrete 

 atoms, or the elementary but con- 

 tinuous portions of matter. Mathe- 

 matically expressed, this means that 

 the relations in question present 

 themselves in the form of differen- 

 tial equations, and that the solution 

 of them consists in finding func- 

 tions of finite (observable) quanti- 

 ties which satisfy the special con- 

 ditions. A comparatively small 

 number of differential equations 

 has thus been found empirically 

 to embrace very large and appar- 

 ently widely separated classes of 

 physical phenomena, suggesting 

 physical relations between those 

 phenomena which might otherwise 

 have remained unnoticed. The 

 physicist or astronomer thus hands 

 over his problems to the mathe- 

 matician, who has either to in- 

 tegrate the differential equations, 

 or, where this is not possible, at 

 least to infer the properties of the 

 functions which would satisfy them 

 in fact, the differential equation 

 becomes a definition of the function 

 or mathematical relation. In con- 

 sequence of this the theory of 

 differential equations is, as Sophus 

 Lie has said, by far the most 

 important branch of mathematics. 



