300 Prof. Challis on the Hydrodynamical Theory of Vibrations, 



disturbance. This, in fact, is done by means of the equation 

 (5) in art. 21, which applies accurately to the motion along an 

 axis of free motion. From the particular solution of this equation 

 obtained in articles 22 and 23, it may be inferred that, however far 

 the approximation be carried, and consequently whatever be the 

 magnitude of the motion along the axis, the rate of propagation 

 of the velocity and condensation is absolutely uniform, and such 

 that no change of the type of the undulations is produced by the 

 transmission. 



46. But it was found in articles 33 and 34 that the equation 

 (5) might be satisfied to the first approximation by supposing (f> 

 to be equal to the sum of an unlimited number of terms such as 

 m sin q (z — a } t-\-c), and that, in order to satisfy the conditions of 

 any given disturbance, it was necessary to express (j> in this man- 

 ner; which analytical circumstance indicates that the motion is 

 of a composite character independently of the mode of disturb- 

 ance. From this first approximation we might proceed to the 

 next by assuming that 



and substituting this value in (5), omitting the last term of the 

 equation. It would then appear that the equation might be 

 satisfied by supposing a/t to consist of a series of sines of circular 

 arcs with constant coefficients of the second order with respect 

 to m. Also if the values of X in the series of the first order be 

 restricted within limits not far apart (as is the case in the Theory 

 of Light), those in the series of the next order will be either con- 

 siderably greater or considerably less. The same thing will happen 

 if the approximation be carried still further. On this account, as 

 w r ell as because ^r is a quantity of a superior order, it is allow- 

 able to omit it in the theoretical explanations of phenomena of 

 light. With respect to the terms of the first order, it may be 

 remarked that the considerations in art. 45 seem to justify the 

 inference that each of the components which they represent is 

 propagated with a velocity absolutely uniform ; but as the expres- 

 sion for that velocity involves m (art. 24), it must be admitted 

 (unless m be the same for all) that the different components are 

 propagated with somewhat different uniform velocities. 



47. 1 have already adverted (art. 1 7) to the equations deduced 

 from the received principles of hydrodynamics for the case of 

 motion in straight lines perpendicular to a plane and propagated 

 in the positive direction, viz. 



w = a Nap. log p =f(z — [a + w) i), 



from which it may be inferred that points of no velocity and con- 

 densation may at the same time be points of maximum velocity 

 and condensation. The interpretation put upon these equations 



