252 Prof. Challis on the Rectilinear Transmission 



Since f is a function of r — that is, of (# 2 +y 2 )* — we have 



~-= -, -—= - ; and hence, for any point on the axis, u = and 

 da r' dy r J r ' 



D=0, as is required by the foregoing supposition. Also, be- 



cause for every point on the axis -*-=(), and -=- = 0, it follows 

 J r dec ' dy ' 



that / has at each point a maximum or minimum value, 



which, in the absence of determining conditions, we may 



assume to be unity. Hence the velocity w along the axis 



will be expressed by — . 



From these results, taken in conjunction with the first and 

 second general hydrodynamical equations, we might proceed 

 to obtain actual values, expressed in series, of the quantities / 

 and <f>, from which the motions along and parallel to the axis, 

 and in the directions transverse to the axis, might be inferred. 

 It is not necessary to introduce here these investigations, since 

 I have already given them at length in arts. 20-28 of a com- 

 munication, " on the Mathematical Theory of the Vibrations 

 of an Elastic Fluid,'"' inserted in the Philosophical Magazine 

 for August 1862, and also in my work on the Principles of 

 Mathematics and Physics, pp. 201-211. It is, at present, 

 chiefly of importance to remark that, as I have been careful 

 to point out, all this investigation is prior to the supposition 

 of any arbitrary method of putting the fluid in motion, and 

 consequently has exclusive reference to laws of the mutual 

 action of the parts of the fluid irrespective of arbitrary con- 

 ditions ; or, as I have said in the present communication, the 

 motion is free, and spontaneous as to its laws. 



Proceeding now to cases of arbitrary disturbance, it is first 

 to be stated that, as the circumstances of spontaneous motion 

 were determined prior to any discussion of arbitrarily imposed 

 motion, the treatment of the latter must take into account the 

 laws obtained relative to spontaneous motion. Accordingly 

 the result of any arbitrary disturbance must be supposed to 

 be a composition of spontaneous motions unlimited as to 

 number and directions, and originating at the place of dis- 

 turbance. The component motions may be designated as 

 elementary motions, which, at the same time that they satisfy 

 by their composition the given conditions of the disturbance 

 severally obey the laws of spontaneous motion obtained as 

 above stated. They are capable of this application on account 

 of their being derived from linear differential equations with 

 constant coefficients. Since the position of the point P was 

 taken ad libitum, what has been argued relative to the resulting 



