Vibration of a System in its Gravest Mode. 567, 



When the determinantal equation is fully expressed, the 

 smallest root, or indeed any other root, can be found by the 

 ordinary processes of successive approximation. In many of 

 the most interesting cases, however, the number of coordi- 

 nates is infinite, and the inclusion of even a moderate number 

 of them in the expressions for T and V would lead to 

 laborious calculations. We may then avail ourselves of the 

 following method of approximating to the value of the 

 smallest root. 



The method is founded upon the principle* that the 

 introduction of a constraint can never lower, and must in 

 general raise, the frequency of any mode of a vibrating 

 system. The first constraint that we impose is the evan- 

 escence of one coordinate, say the last. The lowest fre- 

 quency of the system thus constrained is higher than the 

 lowest frequency of the unconstrained system. Next impose 

 as an additional constraint the evanescence of the last co- 

 ordinate but one. The lowest frequency is again raised. 

 If we continue this process until only one coordinate is left 

 free to vary, we obtain a series of continually increasing 

 quantities as the lowest frequencies of the various systems. 

 Or if we contemplate the operations in the reverse order, we 

 obtain a series of decreasing quantities ending in the precise 

 quantity sought. The first of the series, resulting from the 

 sole variation of the first coordinate, is given by an equation 

 of the first degree, viz. A— p 2 L = 0. The second is the lower 

 root of the determinant (2) of the second order. The third 

 is the lowest root of a determinant of the third order formed 

 by the addition of one row and one column to (2), and so on. 

 This series of quantities ma}^ accordingly be regarded as 

 successive approximations to the value required. Each is 

 nearer than its predecessor to the truth, and all (except of 

 course the last its-elf) are too high. 



The practical success of the method must depend upon the 

 choice of coordinates and of the order in which they are 

 employed. The object is so to arrange matters tbat the 

 variation of the first two or three coordinates shall allow a 

 good approximation to the actual mode of vibration. 



The example by which I propose to illustrate the method 

 is one already considered by Prof. Lamb. It is that of the 

 transverse vibration of a liquid mass contained in a horizontal 

 cylindrical vessel, and of such quantity that the free surface 

 contains the axis of the cylinder (r = 0). If we measure 6 

 vertically downwards, the fluid is limited by >' = 0, r = e, and 

 * 'Theory of Sound,' §§88, 89. 

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