42 Prof. L. Boltzmann on the 



conceptions which before had played a part in Analysis only. 

 From this endeavour towards clearness arose the graphical 

 representations of the fundamental conceptions of mechanics 

 in Maxwell's ' Matter and Motion/ the geometrical repre- 

 sentation of the superposition of two sine motions, all the 

 illustrations due to the theory of quaternions : thus, the 

 geometrical interpretation of the symbol 



d?_ L d?_ <P 

 dx 2 + df + dz* 



A =^ + ^72 + ^-* 



There was another matter. The most surprising and 

 far-reaching analogies were seen to exist between natural 

 phenomena which were apparently quite unlike. Nature 

 seemed in a certain sense to have built up the most diver- 

 sified things after exactly the same pattern ; as the analyst 

 dryly observes, the same differential equations hold for the 

 most diversified phenomena. 



Thus the conductivity of heat, diffusion, and the propa- 

 gation of electricity in conductors takes place according to 

 the same laws. The same equation may be considered as the 

 solution of a problem in hydrodynamics or in the theory of 

 potential. The theory of vortices in fluids as well as those 

 of the friction of gases exhibits the most surprising analogy 

 with that of electromagnetism, &c, &c. Compare on this 

 point Maxwell, Scientific Papers,' vol. i. p. 156. 



Maxwell also, when he undertook the mathematical treat- 

 ment of Faraday's conceptions, was from the very outset 

 impelled by their influences into a new path. Thomson f 

 had already pointed out a series of analogies between pro- 

 blems in the theory of elasticity and those of electromag- 

 netism. In his first paper on Electricity Maxwell t explained 

 that it was not his intention to propound a theory of elec- 

 tricity ; that is, that he himself did not believe in the reality 

 of the incompressible fluids and of the resistances which 

 he there assumes, but that he simply intends to give a 



* Maxwell, 'Treatise on Electricity and Magnetism/ 1873, vol. i. art. 29, 

 nature of the operator v and v 2 ; this was also afterwards observed by 

 others. Mach on M. Guebhard's representation of Equipotential Curves, 

 Wien. Sitzuiigsbericht, vol. lxxxvi. p. 8, 1882. Compare also Riemann, 

 " Electricitat und Magnetism," Wied, Beiblcitter, vol. vii. p. 10 ; Comptes 

 Rendus, vol. xcv. p. 479. 



f Cambridge and Dublin Math. Journal, 1847; Math, and Phys. 

 Papers, vol. i. 



\ Maxwell, on Faraday's Lines of Force. Cambridge Phil. Trans, vol. x ; 

 Scient. Papers, vol. i. p. 157. 



