Notices respecting New Books. 463 



The seventeenth and eighteenth chapters are almost wholly oc- 

 cupied with contributions made by the author to the Theory of 

 Sound. Thus the seventeenth chapter does, indeed, begin with a 

 long extract from Professor Stokes's paper, " On the Communication 

 of Vibrations from a Vibrating Body to a Surrounding Gas," in 

 which he applies his determination of the complete value of \p (the 

 symbol which represents a disturbance propagated wholly outwards) 

 to the explanation of " a remarkable experiment by Leslie, according 

 to which it appeared that the sound of a bell vibrating in a partially 

 exhausted receiver is diminished by the introduction of hydrogen" 

 (vol. ii. p. 207). The explanation of this seemingly paradoxical 

 phenomenon, it may be remarked, had escaped the penetration of 

 Sir J. Herschell, who " thought that the mixture of two gases 

 tending to propagate a sound with different velocities might pro- 

 duce a confusion resulting in a rapid stifling of the sound" (p. 214, 

 vol. ii.). So far the contents of the chapter are due to Professor 

 Stokes ; the remainder is taken from two papers by the author 

 published in the ' Proceedings of the Mathematical Society ' — " On 

 the Vibrations of a Gras contained within a Rigid Spherical Enve- 

 lope," and an " Investigation of the Disturbance produced by a 

 Spherical obstacle on the Waves of ^ound." 



The eighteenth chapter contains a discussion of considerable in- 

 terest from a mathematician's point of view, viz. " a Proof of La- 

 place's Expansion for a Function which is Arbitrary at every point 

 of a Spherical Surface." But to put this in a proper light we must 

 look back to vol. i., where a proof is given of Fourier's Series. The 

 method adopted may be indicated as follows : — The author first 

 considers the motion of a vibrating string when the ends are 

 not absolutely fixed — a state of things which he represents by sup- 

 posing a mass (M), treated as unextended in space, attached to 

 each end and acted on by a spring (fi) towards the position of equi- 

 librium, — and then particularizes his solution in two ways, first, by 

 supposing M=0 and /ul=oo , so that the ends of the string are fast; 

 secondly, by supposing that both fi and M are zero, a case which 

 might be represented by supposing the ends of the string capable 

 of sliding on two smooth rails perpendicular to its length. From 

 the results thus obtained Fourier's Theorem is shown to follow. 

 In connexion with this proof, the author remarks: — ' f So much 

 stress is often laid on special proofs of Fourier's and Laplace's 

 Series, that the student is apt to acquire too contracted a view of 

 the nature of those important results of analysis " (p. 159, vol. i.); 

 and he adds, in a note, that " the best system for proving Fourier's 

 Theorem from Dynamical considerations is an endless chain stretched 

 round a smooth cylinder, or a thin re-entrant column of air inclosed 

 in a ring-shaped tube" (p. 160, vol. i.). 



It will be observed that the remark above quoted implies a 

 promise of a similar discussion of Laplace's Series ; and this is ful- 

 filled in chap, xviii. The " system " adopted is that of a thin sphe- 

 rical sheet of air. In chap, xvii., as we have seen, there is a 

 discussion of the vibrations of a gas contained within a rigid sphe- 



