Prof. Challis on the Principles of Hydrodynamics. 99 



this case to be satisfied, one of the arbitrary functions must be 

 made to disappear. Let the function r = 0. Then 



^j_ f{h-Kat) f{h-Kat) 

 ^~ h h^ 



f{h—Kat) 

 h 



It is again to be remarked, that as these equations contain arbi- 

 trary functions, they can apply only to the arbitrary disturbance, 

 the denominators having no reference to the law of variation of 

 the velocity and condensation with the distance, but to the pro- 

 portion of the two parts of which the velocity, each moment that 

 it is impressed, consists. When V is given as a function of tj 

 the two parts of the velocity are found by the solution of a 

 common differential equation, of which the variables are the 

 function / and the time t. I have given instances of this process 

 in a paper on the Motion of a small Sphere in an elastic medium 

 (Cambridge Philosophical Transactions, vol. vii. part 3). After 

 thus finding /, and by consequence /, the initial condensation a- 

 becomes a known function of t. The initial relation between V 

 and o" is 



which agrees with the general relation we have already obtained 

 between the velocity and condensation when uniform propagation 

 takes place in a rigid tube. The condensation a and the part 



of the velocity — — 7 -, are propagated, after the first instant 



of the generation of the condensation, with the uniform velocity 

 Ka, varying at the same time with the distance according to the 

 law of the inverse square, as proved in Prop. XII. The other 



part of the velocity, —'^-^^ — p — '■-, being accompanied with no 



condensation, or at least with none of the order considered in 

 this investigation, is transmitted instantaneously, varying with 

 the distance inversely as the square of the distance as in an 

 incompressible fluid, and existing only so long as the disturbance 

 is going on. It would seem that the initial generation of con- 

 densation is not possible without this accompanying velocity, 

 which is so much the less as the distance of the place of disturb- 

 ance from the centre is greater. The proposed problem has thus 

 been completely solved on the principles of this theory. 



For the purpose of confirming the foregoing reasoning I pro- 

 ceed to give another solution, which to a certain extent is appli- 

 cable to the same problem. It is evident that if an unlimited 



