Prof. Challis on Hydrodynamics, 213 



for w and a will have to be taken into account. For the vibra- 

 tions of a very elastic medium it is usually required to take 

 account only of the first term of each series. 



Another remark relating to the particular solution of (a) may 

 be suitably added here. When the approximations to the values 

 of w and a are restricted to the second order, the value of a l is 



given with sufficient exactness by the equation a^ — a 2 -^-^- 



47r 2 a 2 

 Hence we have «j 2 =/c 2 a 2 and b 2 = q' 2 a 2 (/c 2 — 1)= -— — (/c 2 — 1). 



Consequently, as the differential equation contains Z>, the solu- 

 tion is true only for a single value of A. It is, however, not the 

 less proper for making known those laws of the motion of the 

 fiuid which are independent of arbitrary circumstances. In fact, 

 the foregoing values of w and a not only indicate the law of 

 uniformity of propagation, and the law of vibration expressed by 

 a circular function, but also show that the principal vibrations 

 are accompanied by others of less magnitude, severally expressed 

 by the terms of a convergent series of circular functions. Al- 

 though the mathematical theories of sound and light usually take 

 account only of the first terms of the values of w and o-, the 

 movements which the other terms represent coexist with those 

 that are principal, and in the case of sound have been actually 

 recognized as producing the harmonics accompanying a musical 

 note. Also the proof that the rate of propagation is exactly 

 uniform depends on the possibility of expressing w and a by these 

 series. 



After the foregoing preliminaries, we may next consider in 

 what manner the values of w and <r are applicable in given cases 

 of motion. The reasoning will at first be confined to approxi- 

 mations of the first order ; and the problem of rectilinear motion 

 perpendicular to a plane, which has been already referred to, will 

 be selected for a first example. I have given a solution of this pro- 

 blem in arts. 32-36 ; and the principles of composite motion, that 

 is, of motion compounded of the motions defined by the first terms 

 of the expressions for w and or, and of accompanying transverse 

 motions, are fully exhibited as far as regards this instance. On 

 reviewing the reasoning, I find nothing that requires correction ; 

 it is for the purpose of adding a supplementary remark which is 

 of considerable importance that I have occasion to advert to that 

 solution. In art. 34 reasons are given for asserting that the 

 successive values of the function Y(z— /cat), the form of which is 

 arbitrary, may be expressed as nearly as we please by the sum 



of such terms as m sin— (z—fcat-\-c) } the number of terms and 



