View of the Undulatory Theory of Light. 191 



Then for the points through which these planes pass, we shall 

 have constantly 



« = w(0), and -^ = /2^(0); (48.) 



or, what is the same thinp;, from (36.) (37.)j 

 « = doA + eoB+foCand^=it/2(goA + hoB + ioC) (49.) 

 And for the planes bounding the waves successively, 



« = .(i-) ^=^n.'(L)., (50.) 



or, what is the same thing, 



« = -[doA + eoB + foC] 



and ^= -^/2(g^A + hoB + ioC] (51.) 



Further, the velocity of the propagation of a plane wave, 

 or, in other words, the velocity of the displacement of the 

 plane (47.) measured perpendicular to it, will he constant by 

 virtue of the formula (47.), and represented by ft. 



If we suppose the functions such as to fulfill the same con- 

 ditions as those of (42.) with only the difference of the sign, or 

 11(g) = -/2^'(^), (52.) 



the same considerations readily show that we should have 



8 = m (g-nt), (53.) 



and by consequence, in the same way as before, 



Ap = SlAt, (54.) 



The inference, then, will here be that the motion of m is 

 immediately transmitted to molecules on the positive side, the 

 velocity being still the positive constant ft. 



It may also be observed in either case, that the formula 

 which determines 5 in functions of 1c for a given direction of 

 the plane (16.), will also determine T or /2 in functions of/. 



If, however, the functions U (g) and m [q) be such that the 

 condition (42.) is not fulfilled either with the positive or ne- 

 gative sign, then we cannot proceed to determine the value 

 of a by the conditions involved in the former investigation ; 

 that is, it will follow that the formula (39.), or the three si- 

 milar formulas involving the three values of 5, will not enable 

 us to determine the nature and conditions of the three displace- 

 ments in directions parallel to the axes of the ellipsoid. But 

 we may consider the value of s as representing a motion pro- 

 duced by the composition of six motions (three on each side 

 of the given plane), each corresponding to that represented 

 by the equations (43.) and (53.)? according to their signs. 



The plane waves corresponding to each of these six mo- 



