Notices respecting New Books. 67 



Por the purpose of giving some notion of the extent to which 

 these subjects are treated, we will indicate briefly the contents of 

 one chapter ; and for this purpose we will take the last, viz. that on 

 Vibrations of Plates — a plate being a thin solid " of uniform iso- 

 tropic material and constant thickness " (p. 293). The general 

 expressions in the case of such a plate are first investigated for 

 the potential energy of each unit of area, and for its variation 

 from which the equation for the motion of the plate at any one 

 point is found, and then the equations of condition arising from the 

 state of its boundary, whether free, clamped, or supported. In 

 subjects of this kind, however, the difficulty only begins when the 

 general equations have been formed ; and accordingly the next step 

 is to modify them to suit the case of a circular plate and to integrate 

 them when thus modified. The results obtained are compared — 

 both in respect to the principal tones, and the nodal lines — with 

 the results of observation ; and a sketch is given of the history of 

 the problem. Two other cases are also discussed. The first is 

 that of a rectangular plate whose edges are free (the case in which 

 the edges are supported being but briefly noticed) ; but in this 

 case the mathematical difficulties necessitate the supposition that 

 p = ; i. e. the lateral contraction is assumed to be evanescent 

 in comparison with the longitudinal extension. The results ob- 

 tained on this supposition as to nodal lines and principal tones are 

 found to admit of pretty close comparison with observations on a 

 square plate. The second case is that of a cylinder or ring. 



This statement will, perhaps, serve to convey some notion of our 

 author's treatment of the several special systems of vibrating bodies. 

 Of the contents of Chapters 4 and 5, which treat of vibrating sys- 

 tems in general, it is not easy to write without going into details 

 such as our limits will not allow. Partly this is due to the ex- 

 treme generality of the statements : e. g. such a statement as this — 

 A force of any type acting alone produces in a system a displace- 

 ment of a second type from the zero configuration equal to a dis- 

 placement of the first type due to the action of an equal force of 

 the second type — is scarcely intelligible apart from the reasoning 

 by which it is proved, though a particular case mentioned by way 

 of illustration is plain enough : — " If A and B be two points of a 

 rod supported horizontally in any manner, the vertical deflection at 

 A when a weight W is attached at B, is the same as the deflection 

 at B when W is applied at A " (p. 69). So, again, " Young's 

 Theorem" (p. 144) is perfectly intelligible as a simple statement, 

 but in its generalized form (p. 99) it is almost unintelligible without 

 its context. 



One important point in these chapters maybe mentioned, viz. 

 the introduction into the Equations of Motion of a function (P) 

 called the " dissipation function," to represent the forces arising 

 from friction and viscosity, P being a homogeneous quadratic 

 function of the velocities. The author, it must be added, has done 

 every thing that could well be expected to smooth down the asperi- 

 ties of a very difficult subject, both in the way of illustration and 



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