Prof. Challis on the Dispersion of Light. 459 



motion is always and everywhere consistent with the principle of 

 constancy of mass. The equation deduced from the above prin- 

 ciple of continuity is 



from which we may, in the first place, infer that, as the complete 

 value of yfr will contain arbitrary functions of the coordinates 

 and the time, there may be an unlimited number of ways in 

 which udx-{-vdy + wdz can be made integrable by a factor, the 

 particular factor in each case being determined by the arbitrary 

 circumstances of the motion. This inference is true whether 

 the motion be large or small, and whether it be vibratory or not. 

 At the same time, as is indicated by pure analysis, if udx + vdy 

 + wdz be integrable without a factor, the quantity \ must be 

 supposed to be generally a function of \jr and t. This analytical 

 circumstance must have its counterpart in the motion of the fluid. 

 In fact I have proved that when that supposition relative to X is 

 introduced into the foregoing equation, the consequent value of 

 the function -\fr shows that the motion is rectilinear. As this 

 result has been reached irrespectively of any arbitrary conditions 

 that might produce rectilinear motion, we may conclude that the 

 rectilinearity is due to the mutual action of the parts of the 

 fluid, and that, so far as that action is unconstrained, there exist, 

 under all circumstances of disturbance, straight lines, or axes, 

 along which the motion takes place. But we have no right to 

 conclude that the motion is wholly, or necessarily rectilinear. 



Again, from the three approximate dynamical equations ad- 

 duced at the commencement of this reasoning, I have demon- 

 strated generally that if udx + vdy + wdz be integrable for small 

 motions, the motion cannot be independent of the time, but must 

 consist of small vibratory motions. Here also a characteristic of 

 the motion has been indicated by the analysis antecedently to any 

 supposed conditions of the motion ; and on this account we are 

 entitled to conclude that vibratory motion results generally from 

 the mutual action of the parts of the fluid. But we have no right 

 to say that in every arbitrary case of small vibratory motion 

 udx + vdy + wdz is an exact differential. This assertion, which 

 is assumed to be true in the ordinary treatment of hydrodyna- 

 mical questions by two general equations, is contradicted by the 

 third one above, which, as we have already argued, proves gene- 

 rally, and prior to any consideration of vibratory motion, that that 

 differential equation becomes an exact differential by being mul- 

 tiplied by a factor, the value of which depends in each particular 

 case on the arbitrary circumstances of the motion. 



Since from the same three approximate differential equations, 



