1 10 Prof. Challis on the general Differential 



between the first and second surfaces, greater or less than that 

 which would exist in the same space in the quiescent state of the 

 fluid, is changed to the quantity between the second and third 

 in the time 8t such that Sr = a'8t, a' being constant. The exact 

 relation between V and p, and the laws of their variation result- 

 ing from these conditions, are found by the integration of equa- 

 tion (4) to be given by the equations 



, ¥{r-a't + c) _ 



If p = 1 + a, and cr be very small, we have very nearly 

 ^Jjr-Jt + c) = ^ 

 r* 

 Now, if the supposed conditions were true, a solitary wave, either 

 of condensation, or of rarefaction, and of constant breadth, might 

 be propagated with the uniform rate a! from a centre \ and the 

 relation between the velocity and density, and the laws of their 

 variation with the distance from the centre, would be correctly 

 given by the above results. But when the problem of propagated 

 motion from a centre is solved for small motions, after taking 

 account of the first general equation, the results are quite differ- 

 ent from those above. Consequently it must be concluded that 

 the propagation of a solitary wave is not possible. This conclu- 

 sion involves another, namely, that the variation of the conden- 

 sation from point to point at a given time cannot be expressed 

 by a discontinuous function ; for if that were the case, the possi- 

 bility of the uniform propagation of a solitary wave would neces- 

 sarily follow. I mention these conclusions the rather because 

 the progress of my hydrodynamical researches was for a long 

 time retarded by the misconception, that results obtained from 

 the equation (4) combined with the principle of discontinuity 

 were necessarily true, and general in their application. The 

 correction of this error is given in an article " On the Central 

 Motion of an Elastic Fluid/' contained in the Philosophical 

 Magazine for January 1859. The principle of the discontinuity 

 of the arbitrary functions can at least have no application prior 

 to the consideration of particular cases of disturbance. 



11. We may now proceed to the revision of Prop. VII. in the 

 Philosophical Magazine for March 1851 (p. 232). The object of 

 this proposition is to trace the consequences of assuming X to be 

 a function of ty and t in the general equation (3), which is, in 

 fact, to assume that udx + vdy + wdz is integrable without a fac- 

 tor. Now this is a possible analytical circumstance of a general 

 character, and, being such, it corresponds, according to the first 

 of the three preliminary remarks in art. 9, to some general cir- 

 cumstance of the motion which is independent of particular 



