98 Prof. Challis on the Principles of Hydrodynamics. 



centre, which condition could not be introduced without error 

 into equations applicable only to free motion. The resulting 

 differential equations and their solution will evidently be the 

 same as those obtained in the ordinary mode of treating this 

 problem, with the difference only that Ka takes the place of a. 

 Hence we have at once 



R ^ R 



f(R-Kat) Yjli-}- Kat) 



__ f(R-Kai) ¥'{U-\-Kat) 

 ^aa jg . ^ . 



It is important to remark that these equations, containing arbi- 

 trary functions, apply exclusively to the given arbitrary disturb- 

 ance. For instance, when ^=0, let V=0, and let the values of <r 

 be given through an arbitrary extent along the tube. Then since 



0= E(/(R) + P(R)) - (yiR) + F(R)) 

 for every value of R through that extent, we must have 



/(R) + r(R) = and /(R) + F(R) = 0. 

 Hence 2/(R) 



it 



The function /(R) is thus determined, and F(R) = ~/(R). If, 

 therefore, 



Vi- ^ ana Va---^-, 



we have Vj + VgrsO, and 



Kaa __^ Kaa- 



V ^^ V 



These two last equations show that the motion at the first instant 

 resolves itself into equal motions, which satisfy the conditions 

 of uniform propagation in opposite directions. The velocities 



— iTe- aiid 4t^^ destroy each other at the first instant, and 



not being accompanied by condensation, give rise to no pro- 

 pagation. The velocity and condensation after the first instant 

 at any point may be inferred from the uniformity of propagation, 

 and from the law of variation according to the inverse square of 

 R, as shown in Proposition XII. 



Next let the velocity be given at a given distance h from the 

 centre during an arbitrary interval, and let it be required to find 

 the resulting condensation. As there is only one condition in 



