a 
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. “M. Young,’’ he says, “ 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.”’ 
And on the same point, Dr. Whewell mentions from personal infor- 
mation, that “ 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 undulatory 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.’’—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 
da dat 
where ¢ is the time, 7, the distance along a given axis, and wu, the dis- 
placement corresponding to the time, ¢; c,a constant. The solution 
of this equation is easily seen to be the wave function. 
wu == sin (nt — kx) 
