332 Prof. Challis on the Undulatory Theory of Light. 

 Consequently 



«;=:-?^cos(2\/^ + c')sin^^-^ + c), 



A A 



— 2tt 



(o= —2m vVo sul ( 3 V<^ + c ') cos — - (z—fcat + c), 



Hence it follows that w= — . Suppose now another series of 



K 



waves, exactly equal to the first, to he propagated in the contrary 

 direction ; and let z be the coordinate of a position at which 

 the velocity parallel to the axis is constantly zero. Then if in 

 general z = z -t-l, and if w', co f , and cr' be respectively the result- 

 ing velocities and condensation, we shall have 



«/=- ?^cos(2 ^eh + e')(sm^ (l-mt) + sin 2 ^(l+mt)y 

 o)'= —2m \fef sm (2 Veh + c f )[cos-^(l—Kat) + cos -^ (I +Kat)\ 



A# \ A A / 



From the first and second of these equations it appears that 



g>' ^/g\ - 27rZ 



— = . tan (2 v eh + c) cot -r — 



Let now the distance r apply to positions at which the trans- 

 verse velocity is always zero ; and in order to get rid of the ne- 

 gative signs, and to avoid double signs, let d = (2n + IJ-tt. Then 

 supposing h and / to represent very small equal distances from a 

 point (r , z ), where the velocity is constantly zero, we have 

 ultimately 



ft>o_ s/"e\ tan 2 s/~e h ___ ^ 2 



w f o ir * A 2tt/ ~ 7T 2 



tan— - 

 A 



This result informs us that the changes of condensation produced 

 by the flow of the fluid to or from any point of no velocity are 

 due to the longitudinal and the transverse motions in a constant 

 ratio. It hence follows that the changes of condensation at any 

 given point of a single series of waves are due at each instant to 

 the longitudinal and transverse motions in the same ratio. In 



