Prof. Challis on the Hydrodynamical Theory of Vibrations, 295 



this is equally prevented whether the infinitely small transverse 

 sections be uniform, or variable in the maimer just mentioned, 

 the same equation applies in all cases. Let us suppose, for 

 instance, that the disturbance of the fluid is such that the motion 

 is constrained to take place in all directions from a centre, and 

 so as to be a function of the distance from the centre. It will 

 be sufficient in this case to consider the motion in a slender 

 pyramidal tube bounded by planes passing through the centre ; 

 and if V and a be the velocity and condensation at the distance 

 r from the centre, we have by the equation above, 



2 2 d<r dW 



dr at 



Also by the equation of constancy of mass, 



da ^V _2V 



dt dr r 



These two equations conduct to the usual solution of the pro- 

 blem of central motion, with the exception that ica takes the 

 place of a. It must, however, be borne in mind that the fore- 

 going investigation rests on the law of composite motion, the 

 components being, as in the case of motion perpendicular to a 

 plane, vibrations expressed by circular functions. By means of 

 this composition it is possible to explain how the condensation 

 varies inversely as the distance, and not inversely as the square 

 of the distance, from the centre. It does not fall within my pre- 

 sent purpose to discuss at greater length the case of central motion. 



38. Not only in the cases above considered of parallel and 

 central rectilinear motions, but in every instance of disturbance 

 of the fluid, however produced, both initially and subsequently 

 the velocity and condensation must be regarded as compounded 

 of velocities and condensations defined by the functions <j> and/. 

 This follows from the circumstance that the forms and values of 

 these functions were determined antecedently to the considera- 

 tion of any case of disturbance. To satisfy the conditions of a 

 given disturbance, the number and directions of the axes of rec- 

 tilinear motion, and the values of m, \, and c, are at disposal. 



39. According to the determinations that have been made of 

 the forms of <fi and f, the vibrations which take place relative to 

 a single axis of rectilinear motion, which, as being independent 

 of the mode of disturbance, I have called for distinction free 

 motions, are such as follow. The motion along and parallel to 

 the axis is represented to the first approximation by 



w/sin— - (/cat — -z + c) ; 



