502 REPORT— 1902. 



these surfaces being the same for all the rays. The proof he gives of this theorem 

 is so simple that it now seems almost axiomatic ; but it is curious that Malus, who 

 had made the laws of Light his special study, though he suspected that the 

 theorem ought to hold, yet found himself unable to establish it. 



Hamilton, now considering the length of the path to any point as a function 

 of the coordinates of that point, and denoting this function by V, proves that V 

 satisfies a simple partial differential equation of the first order and proceeds to 

 show the important part the function Y plays in the theory. 



He goes on to prove generally that if we are dealing, not with right lines, 



that is, with paths, for which as between any two points Irf*- is a minimum, 



but with curved paths for which LifZ.5'isa minimum (where ^ is a function of 



the coordinates), and a system of such paths be drawn through a given point, O, 

 the system of surfaces V - const, will still cut all the paths at right angles. If we 

 adopt the emission theory of Light, and we take for n the velocity of Light, V 

 becomes ' the Action,' and the minimum property which the paths satisfy is the 

 principle of ' Least Action.' If, on the other hand, we adopt the undulatorv theory, 

 and we take for ft the reciprocal of the velocity, the minimum property becomes 

 the principle of ' Least Time.' Thus Hamilton shows that, by altering the signifi- 

 cance of fji, his method applies to either theory. 



Introducing the further conception that /x depends, not only on the coordinates 

 of the point, but also on the direction-angles of the ray, be is able to apply his 

 reasoning to rays passing through a crystal. Pie gives by his method a new and 

 interesting proof of the equation of Fresnel's wave-surface, and arrives at the 

 conclusion, hitherto unnoticed by mathematicians, that this wave-surface possesses 

 four conical cusps and also four special tangent planes, each of which touches the 

 surface, not in one point only, but in an infinite system of points lying in a circle. 

 The physical significance of these theorems is what is known as Conical Refraction, 



Having drawn this inference from his mathematical analysis, Hamilton wrote to 

 his friend Dr. Lloyd and asked him to verify it by actual observation, and 

 accordingly Hamilton's paper in the ' Transactions ' of the A.cademy is accompanied 

 by another from Lloyd describing the beautiful arrangements by which he had 

 succeeded in verifying this remarkable phenomenon in both its varieties. 



This striking instance of scientific prediction naturally made a great sensation 

 at the time, appealing, as it did, to a much larger public than the few select 

 mathematicians who were capable of mastering the elaborate treatise on 

 * Systems of Rays.' 



The experimental skill that was required to obtain these results may be 

 realised from the circumstance that as I have been told the French physicists 

 found themselves unable to repeat the experiment till Lloyd himself went over to 

 Paris with his instruments and showed them the way. 



Hamilton was so well satisfied with the success of his new method in dealing 

 with the problems presented by the propagation of Light that full of enthusiasm 

 he proceeded to apply a generalised form of the same method in the investigations 

 of the motion of any material system, and a paper of his was read before the 

 Royal Society in 1834 with the following title : ' On a general method in Dynamics 

 by which the Study of the Motions of all free systems of attracting or repelling 

 points is reduced to the Search and Differentiation of one Central Relation, or 

 Characteristic Function.' 



To show the importance attached by the most competent judges to Hamilton's 

 work in this field of Theoretical Dynamics, we cannot do better than quote the 

 words of his great German contemporary Jncobi, who afterwards himself added 

 to the new theory such valuable developments. 



Jacobi writes as follows : — ' If a free system of material points is acted on by 

 no other forces than such as arise from their mutual attraction or repulsion, 

 the differential equations of their motion can be represented in a simple manner 

 by means of the partial differential coefficients of a single function of the co- 

 ordinates. Lagrange, who first made this important observation, at the same time 



