on the Hypothesis of Undulations, . . 467 



portions are polarized in planes at right angles to each other. This 

 resolution, it appears (art. 42), applies only to transverse motions 

 immediately contiguous to the axis, — which, however, is all that 

 the theory requires, reasons having been already given for con- 

 cluding that the transverse motions of the primitive ray, so far 

 as they are light-producing, are similarly restricted. 



For a polarized ray, the transverse vibrations of which are 



Sx 2 

 parallel to the plane of xz, f is equal to 1— -=-g- (art. 28), its 



value for a point on the axis being supposed to be unity. Hence 

 the equations which give the velocities and condensation of a ray 

 completely polarized in the plane of xz are the following : — 



. df Smx 2tt , , , 



u = 6-j-= — —cos—- (teat— z + c), 

 T ax ir\ A, 



M /, 8a?*\ 2ir f ... _ s 



W =f/z =m { l --^J Sm Y {Kat - Z + c) > 



and the transverse accelerative force is 



dcr \-6m/cax . 2tt , , . 



— a z ^r— — r-^ — sin -— [teat— z-\- c). 

 dx X 2 X v ; 



From these equations it is evident that the circumstances 

 under which two rays polarized in the same plane, and having 

 coincident axes and the same value of A,, interfere or coalesce, 

 are precisely the same as those already found for two primitive 

 rays. But if the rays be polarized in planes at right angles to 

 each other, different results are obtained. Let the two rays be 

 in all other respects exactly alike, and, first, let their phases be 



the same, or differ by an even multiple of 5. Then we have for 



the transverse velocity (v) in the plane of yz, 



. df Smy 2ir , , x 



v=d>-T- = -—^ cos — (teat— z + c). 

 T ay w\ \ 



Hence the resulting transverse velocity, or (w 2 -f v 2 )*, is 



Smr 2tt . N 



— — cos—- {/cat — z-\ c) : 

 7TA, X x J 



that is, the same as for a primitive ray, the value of m for which 

 is double that for each of the polarized rays. This will also be 

 the case with respect to the resulting values of w, a, and the 

 transverse accelerative force ; so that the compound ray will differ 

 in no respect from a primitive ray. If now the difference of phase 



