210 Prof. Challis on Hydrodynamics. 



inference. For if such a partition be made to separate parts of 

 the fluid in a state of condensation or rarefaction, and at the 

 same time to partake of the motion of the fluid in contact with 

 it, since no assignable force is thus introduced, it is evident that 

 the condensation is only required to satisfy the condition of being 

 equal on the opposite sides of the partition. It is not neces- 

 sary that the changes of condensation from point to point at a 

 given instant on the one side and on the other should be ex- 

 pressed by the same function. Consequently, the partition being 

 conceived to be removed, ordinates drawn to represent the con- 

 densation will have consecutive values, but the directions of the 

 tangents to the line joining their extremities may change per 

 saltum. Hence the motion of a given element is not generally 

 expressible by a single function, but by different functions, in 

 such manner, however, that the velocity always changes gradatim. 

 The motion is in fact analogous to that of a material particle 

 acted upon by a central force, which from time to time changes 

 abruptly both as to law and amount. The path of the particle 

 would in that case consist of portions of different curves so 

 joined together as to have common tangents at the points of 

 junction. From the above considerations it will be seen that 

 the theory of the solitary wave presents a difficulty which it is 

 absolutely necessary to remove before advancing further in these 

 researches. To do this is the object of the following argument. 

 For the purpose of conveniently indicating the course of the 

 reasoning, I shall here cite the equation (5) in art. 21 (August 

 Number), viz. 



b * ' a dz* + fi* +2 fc dJdi + d^ ~d?-°- ■ w 



It will not be necessary to repeat the process by which this 

 equation was obtained, as sufficient details on this head are given 

 in arts. 14-22. The first term and the factor b 2 are due entirely 

 to the use that was made in that process of the third general 

 equation, which is the equation (3) obtained in art. 7. It is 

 known that the exact equation applicable to rectilinear motion 

 perpendicular to a plane, as obtained by means of the first 

 and second general equations, is the equation (a) deprived of 

 its first term. If, therefore, an integral of that equation can 

 be obtained, and be such as to admit of receiving an interpreta- 

 tion consistent with the motion of a fluid, that circumstance 

 might be adduced as an argument against the necessity of a third 

 general equation. An integral is, in fact, obtainable ; but, as I 

 have pointed out in arts. 17 and 47, and have proved elsewhere, 

 it does not admit of being interpreted consistently with the 

 necessary conditions of fluid in motion. This result is pre- 



