where 



436 Dr. G. J. Stoney on Microscopic Vision. 



which last are constants that are determined by the ellipticity 

 of the light of the undulation U, by the position of its ellipse, 

 and by its initial phase. 



Similarly, another undulation V of uniform plane waves 

 travelling in the same direction will be represented by equa- 

 tions 3 and 4 — 



^ = ^sin(^-.T)+« / ), .... (3) 



£'=&' sin (^ (*;*-#) +/3') (4) 



If the undulations U and Y are simultaneously present, the 

 displacements over the plane x and at the time t will be 

 rj + r]' and f+ f ; and these by elementary trigonometry are 

 found to be 



*+Y=Mrin(^ (*-*)+ A), . ... (5) 



?+5'=Nsin( 2 ^(^- t r)+B), . ... (6) 



M 2 =a 2 + a' 2 + 2aa'cos(«-a')> • • • • (?) 



. a sin a + a! sin a! /ox 



tanA= —7 7, (8) 



a cos a + a cos a 



N 2 = & 2 + & /2 + 266'cos(/3-£'), .... (9) 



tanB=7 %— 77 3-, (10) 



6 cos p + 6 cos £r v ' 



Equations (5) and (6) represent an undulation of uniform 

 plane waves travelling in the same direction as U and V. 

 We may call this resultant undulation W. Equations (7), 

 (8), (9), (10) enable us to determine the constants of any one 

 of these three undulations, if we know those of the other two. 

 It appears accordingly that any two of the three undulations 

 U, V, W being given, the third can be found. 



It is an easy inference from this that any number of undu- 

 lations of uniform plane waves of wave-length X, that travel 

 in the same direction, may be combined into a single undula- 

 tion of the same kind travelling in that direction : a proposition 

 of which use was made above in the latter part of § 6 ; of 

 Part I., p. 337. 



29. Of elementary sheafs of beams, and of the single beams 

 which may be substituted for them. — Beams of uniform plane 

 waves may be emitted in any or all directions from the front 



