90 MATHEMATICS 



the labile ether theory developed by Cauchy and 

 Kelvin, and the rotational ether theory of Mac- 

 Cullagh were all efforts of the kind here indicated; 

 they were all successful in some greater or less 

 degree in the representation of the phenomena, 

 and they all stimulated Physicists to further efforts 

 to obtain more minute knowledge of those pheno- 

 mena. Even such an inadequate theory as that 

 of Fresnel led to the very interesting observation 

 by Humphry Lloyd of the phenomenon of conical 

 refraction in crystals, as the result of the prediction 

 by W. Rowan Hamilton that the phenomenon was 

 a necessary consequence of the mathematical fact 

 that Fresnel's wave surface in a biaxal crystal 

 possesses four conical points. 



In the history of Science it is possible to find 

 cases in which the pendency of Mathematics to 

 express itself in forms of ever increasing degrees 

 of abstractness has proved to be of ultimate service 

 in the physical order of ideas. Perhaps the most 

 striking example is to be found in the development 

 of abstract Dynamics. The greatest treatise on 

 this subject which the world has seen is Lagrange's 

 Mec unique Analytique published in 1788. The 

 spirit in which this great work was conceived is 

 best described in the Preface. Lagrange writes : 

 "We have already various treatises on Mechanics, 

 but the plan of this one is entirely new. I intend 

 to reduce the theory of this Science, and the art 

 of solving problems relating to it, to general 



