IN THE INTERIOR OF TRANSPARENT BODIES. 



da 



429 



curring function both of cf> and t ; for — = 



du 

 da 

 di 



, see Art. (33); hence, 



1) 



since a is a recurring function of u and t, — must be so also (at least 



z sin i 



in general), and therefore, since u = BC = ;-; -, flA't) must be a 



° cos (i - yt) r * T ' 



recurring function of <p' and t: the increments of <j>' and £ which make 

 f(<p't) recur being of course extremely small. 



The form of the function f depends on the law of vibration, if we 

 knew f we could determine the course of the ray from the two equations 



sin =/(£'*) sin 0' (1), 



.(2), 



sin <p _ sin <p' 



sin \J/ — sin (» — <£') "" 



the second equation being obtained from the two first equations at 

 the commencement of this Article, substituting for >// its value i — (f>. 



Without knowing the form of f we may conclude from its periodical 

 nature that the equation (1) is satisfied by several different values of <f>, 

 supposing <p given*, (supposing also for a moment that t is constant,) 

 and then from the equation (2) we may obtain a set of corresponding 

 values of >//. 



Hence it follows, that a single homogeneous ray incident on a prism 

 emerges in several different directions at a given instant. 



* This will appear immediately if we construct a curve in which the abscissa AM=.<f>, 

 and its ordinate MP = <p. 



It is evident that the locus of P will be some 

 such undulating curve as is represented in the figure. 

 If, therefore, we give <p one particular value AQ, 

 and draw QS parallel to AM in order to determine 

 the corresponding value or values of <p', we shall ob- 

 tain several different values of <j>. 



Vol. VII. Paet III. 3C 



