214 Prof. Challis on Hydrodijnamics. - 



the values of m, X, and c for each term being arbitrary. Now it 

 has been argued, in the previous part of this communication, 

 that, in consequence of the principle of the easy divisibility of the 

 fluid, the function F may be discontinuous. To satisfy this 

 condition by the composite motion, it is necessary to admit that 

 the constants m, X, c may vary per saltum. Nothing in the an- 

 tecedent reasoning is opposed to this admission, if only the values 

 of the composite velocity and condensation do not varyjoer saltum. 

 From this argument we may draw the conclusion that a solitary 

 wave, such, for instance, as that of which the condensation is 



represented by the values of the function m sin -— — taken from 



X 



a? = to sc= ^, or that of which the (negative) condensation is 



X 

 represented by the values of the same function from x = ~ to 



%=X, may be propagated uniformly in the fluid without under- 

 going alteration. 



I proceed now to the problem of motion tending to or from a 

 centre, with respect to which I have recently seen reason to 

 come to conclusions at variance with the views expressed in art. 

 10. Although the integrations required by this problem are 

 effected without difficulty, the principles appropriate to the inter- 

 pretation of them are not readily discoverable. For the pur- 

 pose of elucidating the subsequent reasoning, I shall first take 

 the simple example of the central motion of an incompres- 

 sible fluid. The velocity (V) being a function of the distance 

 from the centre, udx + vdy + wdz is an exact differential, and may 



be supposed to be (d<p). Hence V== j-. Also the equation of 



d^.rcb 

 constancy of mass becomes * ^ =0, the integration of which 



gives 



The second of these equations proves that the velocity at each 

 instant varies inversely as the square of the distance from the 

 centre. The equation that gives the pressure (p) is 



# d£ 



P ^dt^2dr* U * 

 Hence by substitution, 



As this equation contains two arbitrary functions, two condi- 



