GENERAL THEORY. 213 



It is impossible not to feel astonishment, when we for 

 the first time learn that two rays of light can mutually 



might oscillate in parallel directions transverse to the direction of the 

 ray; though he thought that the longitudinal vibrations might exist 

 also. But he long hesitated to adopt such an idea, regarding it as 

 inexplicable on any dynamical principles. Fresnel independently 

 started the same idea of transverse vibrations, alone; but he was 

 equally reluctant to propose it, on the ground of a similar mechanical 

 difficulty; yet he distinctly .acknowledged Young s priority in the 

 announcement of the general idea. &quot;M. Young,&quot; he says, &quot;more 

 bold in his conjectures and less confiding in the views of geometers, 

 published it before me, though perhaps he thought of it after me. 11 

 And on the same point, Dr. Whewell mentions from personal infor 

 mation, that &quot; Arago was wont to relate that when he and Fresnel 

 had obtained their joint experimental results of the non-interference 

 of oppositely polarized pencils, and that when Fresnel had pointed 

 out that transverse vibrations were the only possible translation of 

 this fact into the undnlatory theory, he himself protested that he had 

 not the courage to publish such a conception; and, accordingly, the 

 second part of the memoir was published in Fresnel s name alone. 

 What renders this more remarkable is, that it occurred when Arago 

 had in his possession the very letter of Young (1818), in which he 

 proposed the same suggestion.&quot; Hist, of Inductive Sciences, ii. 418. 



Fresnel deduced transverse vibrations on dynamical grounds which 

 had been open to some degree of question. But the nature of the 

 relation between the partial differential equation which he gives, and 

 the wave function which is the solution of it, clearly involves no nec 

 essary restriction of the direction of vibration. That equation is of 

 the same general form as that given by Euler*as referring to sound. 

 Such an equation suffices for light considered as homogeneous. It 

 expresses generally the relation of particles in motion, such that 

 if the time and the position of the particles be increased by corre 

 sponding changes, the form of the function will be unaltered, or the 

 motions recur periodically, which constitutes the essential idea of a 

 wave. Its form is generally 



d 2 u d-u 



~dt? == ~ C cW 



where t is the time, x, the distance along a given axis, and u, the dis 

 placement corresponding to the time, t ; c, a constant. The solution 

 of this equation is easily seen to be the wave function. 

 u sin (nt lex) 



