142 Prof. Challis on the Mathematical Theory of 



velocity and density at any point of the axis of rectilinear motion 

 is propagated with the uniform velocity a v Let us introduce 

 this condition into the equation (5) by substituting F(z — a x t) 

 for (/>. Putting for brevity //, for z—a x t and F for F(/a), the 

 result is 



Integrating now to the first approximation, the value obtained 

 for <f> is precisely that given by the very different process in 

 art, 22. This proves that no other form of solution is compa- 

 tible with uniform propagation. It is observable that if the 

 equation (5) had not contained the term 6 2 <£, the integration to 

 the first approximation would have given a uniform velocity of 

 propagation equal to a 3 independently of the form of the func- 

 tion <f>. 



24. In the communication contained in the Philosophical 

 Magazine for December 1852, I have gone through a somewhat 

 intricate process of analytical reasoning to prove that, on the 

 hypothesis of uniform propagation, 



and, by consequence^ that the exact value of is a function of 

 z—a x t. The same result is more simply deduced from the 

 equation (7), which shows that, however far the approximation 

 be carried, F or is a function of z — a x t of specific form. In 

 the communication just referred to, I have integrated that equa- 

 tion to terms inclusive of the third power of m, and obtained the 

 value of a x to terms inclusive of the square of m. Putting for 

 shortness fjJ for z — aj + c, the following are the results : — 



771 O a 771 o / o a 1 \ 



= m cos qp! g^- 1 sin qfj - -gfr [^- - -J cos 3^, (8) 



fll w + p + .v( 3 # + fy o) 



Hence it appears, as was remarked at the end of my communi- 

 cation to the Philosophical Magazine for last April, the rate of 

 propagation is not altogether independent of the maximum velo- 

 city of vibration. 



25. The next proposition, the proof of which is required by 

 the course of the present investigation, may be thus enunciated: — 

 Terms of the first order being alone taken into account, the func- 

 tion udx + vdy + wdz is an exact differential for points at any 

 distance from the axis of rectilinear motion. This proposition 

 may be proved as follows. Let the pressure at the point xyz at 



