46 



SCIENCE. 



[N. S. Vol. XI. No. 263. 



" It must certainly be regarded as a pleas- 

 ing prospect to analysts," he says in his 

 preface, " that at a time when astronomy 

 from the state of perfection to which it has 

 attained, leaves little room for further ap- 

 plications of their art, the rest of the phys- 

 ical sciences should show themselves daily 

 more and more willing to submit to it. * * * 

 Should the present essay tend in any way 

 to facilitate the application of analysis to 

 one of the most interesting of the physical 

 sciences, the author will deem himself amply 

 repaid for any labor he may have bestowed 

 upon it ; and it is hoped the difficulty of 

 the subject will incline mathematicians to 

 read this work with indulgence, more par- 

 ticularly when they are informed that it 

 was written by a young man, who has been 

 obliged to obtain the little knowledge he 

 possesses, at such intervals and by such 

 means, as other indispensable avocations 

 which offer but few opportunities of mental 

 improvement, afforded." Where in the his- 

 tory of science have we a finer instance of 

 that sort of modesty which springs from a 

 knowledge of things ? 



The completion of the potential theory, 

 so far as it depends on the ISTewtonian law 

 of the inverse square of the distance, must 

 be credited to Gauss, though a host of 

 writers has since contributed many valu- 

 able additions in the way of details. Early 

 in the century Gauss had begun the study 

 of the absorbing problems of the day, 

 namely, problems of attractions and repul- 

 sions. The prevailing notion of mathe- 

 matical physicists seems to have been that 

 all mechanical phenomena may be at- 

 tributed to atti'actions and repulsions be- 

 tween the ultimate particles of matter and 

 the ultimate particles of ' fluids' associated 

 with matter. The difficulties of action at a 

 distance, without the aid of an intervening 

 medium, happily, did not trouble them at 

 that time ; for who shall say that their 

 labors would have been more fruitful if 



they had stopped to remove these diffi- 

 culties? Gauss's first memoir in this field 

 relates to the attractions of homogeneous 

 ellipsoidal masses,* and dates from 1813. 

 It was in this memoir that he published a 

 number of the elegant theorems f which 

 are now found in the elementary books on 

 the theory of the potential function. In 

 1829 he published his theory of fluid figures 

 in equilibrium, J and in 1832 there followed 

 one of the most important papers of the 

 century on the intensity of terrestrial 

 magnetic force expressed in what we now 

 call absolute units. § Six years later he 

 published his wonderful theory of the 

 earth's magnetism || and applied it to all 

 existing observational data. This theory 

 is a splendid application of the potential 

 theory, and his entire investigation is one 

 of the most beautiful and useful contribu- 

 tions to mathematical physics of the cen- 

 tury. Well was he qualified, therefore, to 

 complete the theory of the Newtonian 

 potential function in the collection of 

 theorems published in his memoir^ of 



* ' Theoria attractionis corporum sphaeroidicorum 

 elliptioorum homogeneorum,' 1813. See Gauss's 

 Werke, Band V., Gottingen, 1877. 



f Especially the theorem giving the values of the 

 surface integral 



^cos (s, n) . 



P 



^dS, 



where dS is an element of any closed surface, s the 

 distance from dS to any fixed point, and n indicates 

 the normal to the surface at dS. This gave the key 

 to the very important theorem of the surface integral 

 of the normal acceleration, or 





dS. 



t ' Principia generalia theorise figurse fluidorum 

 in statu sequilibrii, ' 1829. Werke, Baud V. 



? ' Intensitas vis maguetic£e terrestris ad mensuram 

 absolutara revocata.' Werke, Band V. 



II AUegemine Theorie des Erdmagnetismus. Werke, 

 Band V. 



1[ ' AUegemine Lehrsiitze in Beziehung ant die im 

 verkehrten Verhiiltnisse das Quadrats der Eutfernung 

 wirkendeu Anziehungs- und Abstossungs-Kriif te. ' 

 Werke, Band V. 



