250 Prof. Challis on the Rectilinear T, 



ransmission 



pressed by general equations, all other related facts of obser- 

 vation admit of being accounted for by mathematical reasoning 

 derived from the equations. The science of Analytical 

 Hydrodynamics must be pronounced to be in a very im- 

 perfect state, unless it be capable of giving a reason for such 

 a patent and general fact as that just mentioned. The argu- 

 ment I am about to adduce is intended to give the required 

 explanation. It will be proper to premise here that this 

 argument has nothing to do with the necessity I have so often 

 insisted upon, of a third general equation for completing the 

 analytical principles of hydrodynamics. 1 still maintain that 

 the principles on which that equation rests, and the process of 

 its deduction, are so simple and direct that they do not admit 

 of being controverted; but they do not come under consider- 

 ation in the present enquiry. 



Supposing the reasoning to apply to a portion of the fluid 

 contained in a straight rectangular tuba of small uniform 

 transverse section extending from A to P, it is evident that 

 if arbitrary motion be communicated to the fluid by a movable 

 diaphragm at A exactly fitting the tube, rectilinear motion will 

 take place in the direction of the axis of the tube. This 

 problem has been discussed by Poisson in the Journal de 

 VEcole Poll/technique, torn, vii., for the case in which p = d 2 p ; 

 and from the mathematical investigation he deduces a par- 

 ticular integral from which he infers (p. 369) that the original 

 disturbance will be transmitted uniformly with a velocity- 

 equal to a. The same problem has been solved by Mr. Earn- 

 shaw, in the Philosophical Transactions for 1860, p. 133, by 

 a process differing only in form from that of Poisson, and 

 conducting to the same results. From this integral I ob- 

 tained in a particular case the absurd conclusion that the 

 same particle of fluid might be at rest and have a maximum 

 motion at the same instant of time (see Philosophical Maga- 

 zine for June 1848, p. 496, and my ' Principles of Mathe- 

 matics and Physics,' p. 195). The reasoning by which this 

 result was reached is so certain that it does not admit of 

 being questioned, and, in fact, has not been directly questioned ; 

 but Mr. Earnshaw has thought good to say, that the wave 

 " will force its way in violation of our equations." This is 

 so strange and inadmissible an assertion that I have a right 

 to regard it as a virtual acknowledgment of the recluctio ad 

 abmrdum above mentioned, the reality of which I shall ac- 

 cordingly take for granted. 



But a reductio ad absurdum is not a result which can be 

 slurred over. It is absolutely necessary for the satisfactory 

 establishment of the principles of any science in which it 



