Prof. Challis on Hydrodynamics. 39 



This reasoning proves that the different sets of vibrations may 

 coexist, and that the resulting compound motion is the sum of 

 the separate motions. It follows also that there may be an un- 

 limited number of axes of motion having their positions in space 

 perfectly arbitrary. 



Again, from the approximate general equations already cited, 

 it is readily shown that 



d*<r _ 2 (d^a d*a d?a\ 

 dt* ~~ a \dx €+ dif + dz*)' 



To obtain this equation, no other condition is required than 

 that the motion be small compared to a. It is not necessary 

 to suppose either that udcc + vdy + wdz is an exact differentiator 

 that a contains no part which is independent of the time. If, 

 however, the equation be applied to vibratory motion, the former 

 of these conditions is, as we have seen, in fact satisfied ; but no 

 necessity exists for the fulfilment of the latter. Now the fore- 

 going series for o-', inasmuch as it contains terms involving the 

 squares of the sines of circular arcs, has a value which is partly 

 independent of the time. But for the reason just adduced, the 

 above linear differential equation is nevertheless applicable to 

 vibratory motion relative to an axis, and by means of it the coex- 

 istence of the condensations of different sets of vibrations may 

 be proved by reasoning analogous to that by which the coexist- 

 ence of the velocities was inferred. It is evident that these con- 

 siderations were necessary to complete the proof of the coexist- 

 ence of vibratory motions relative to axes; which, as obeying 

 laws that are independent of arbitrary circumstances, may for 

 distinction be called free vibrations. 



Let us now make an application of the foregoing principles 

 and results by taking the simple case of motion compelled by 

 arbitrary conditions to be wholly in straight lines perpendicular 

 to a plane. Since arbitrarily impressed motion must in any 

 case be such as to result from the composition of free motions, 

 it may, in this instance, be supposed to be compounded of an 

 unlimited number of free motions having their axes all perpendi- 

 cular to the plane, and distributed in such manner that the trans- 

 verse motions are destroyed. In that case, if Vj be the given 

 arbitrary velocity, so impressed that the propagation of the 

 motion is wholly in the positive direction, and if w be the velo- 

 city relative to any one axis, in the direction parallel to it, and 

 at any distance from it, we shall have 



%=k . .fcS . w] =f{z-fcat + c), 



the form of the function / being determined by the arbitrary 

 disturbance, and 2 . w representing the sum, at a given position, 



