Prof. Challis on some Theorems in Hydrodynamics* 343 



have generally for propagated motion in a prismatic tube, 



V = Kaa == yjr(z — feat + c) . 



By changing the sign of fca, these equations would apply to 

 motion propagated in the contrary direction : and if we now 

 suppose V and <r to represent the velocity and condensation 

 resulting from two coexistent and opposite propagations, we 

 shall have 



V = yfr(z — Kat + c) + % (z + /cat-{-c f ) , 



/ca<T=y}r(z — teat + c) ~%(^ + /cat + c'). 



From these equations we readily obtain by differentiation, 



2 % do- dV 



Ka T 2 + Tt=°> 



by which it appears that the accelerative force of the fluid in a 

 prismatic tube is to that along an axis of free motion in the ratio 

 of K 2 to 1, the numerical value of k being 1*18545. 



From the above result we may proceed to the consideration of 

 motion in a straight tube of arbitrary transverse section, subject, 

 however, to the condition that the boundaries of the tube are 

 inclined at indefinitely small angles to the axis of z. To take 

 account of the effect of the rigidity of the tube on the accelera- 

 tive force of the fluid, we may consider the transverse section 

 constant for a very small space, and the equation last obtained 

 may thus be immediately applied ; and to take account of the 

 variation of transverse section, the usual equation of constancy 

 of mass for small motions is required. Thus the equations 

 generally applicable to constrained motion in straight tubes are 



and 



" dz ! dt - U 



dcr .da.dv dw _ n 

 dt doc dy dz ~~ ' 



In the February Number of the Philosophical Magazine I 

 have applied these two equations to the case of motion in straight 

 lines drawn from a centre, the velocity being assumed to be a 

 function of the distance from the centre, this case being evidently 

 reducible to motion in a straight slender tube. I beg to invite 

 the attention of mathematicians to the reasoning there employed, 

 no intelligible solution of this important problem having hitherto 

 been given on the commonly received principles of hydrodynamics. 



The main object I have had in view in prosecuting these 

 hydrodynamical researches, has been to lay a secure foundation 

 for the undulatory theory of light. The theory which attributes 

 the phenomena of light to the oscillations of the individual atoms 



