of Attractive and Repulsive Forces-. 179 



tance from the plane. As preliminary to the solution of our 

 problem, we have to determine under what hydrodynamical 

 conditions such motion takes place. As long ago as the year 

 1848 I pointed out that the exact solution, according to the 

 usually received principles of hydrodynamics, of the problem 

 of plane-waves conducted to an absurd result, indicative of 

 defect of principle. Subsequently I ascertained that the failure 

 of the reasoning arose from not previously taking into account 

 the primary spontaneous vibrations, which, as already argued 

 in art. 8, it is logically necessary to employ for satisfying 

 the conditions of arbitrarily imposed motion. In the theory 

 of sounds it appeared that the required conditions could be 

 fulfilled by means of sums of spontaneous vibrations (arts. 8 

 and 9). In like manner, as I am about to show, the condi- 

 tions of the problem of plane-waves may be fulfilled by sums 

 of uncompounded aetherial vibrations. It should, however, be 

 noticed that whereas for aerial vibrations the transverse 

 motions might be neglected and the factors /, g, A, &c. each 

 be taken to be unity, this is no longer the case when we have 

 to deal with such small wave-lengths as those which pertain 

 to vibrations of the aether. Considering, however, that in the 

 proposed application of the formulas for the primary vibrations 

 the distance r from the axis will always be very small com- 

 pared with X, and that the constant e varies inversely as X 2 , it 

 will suffice to take / and g each equal to 1 — er 2 . Also, on 



7Tb 



account of the smallness of the factor — * for the aether, it will 



a 2 ; 



not be necessary to include in the expression for yfr (art. 7), 

 the terms beyond that which contains m 2 . These limitations 

 being admitted, we may proceed to find approximate expres- 

 sions, in the case of a set of uncompounded spontaneous vibra- 

 tions, for the direct and transverse velocities and for the 

 condensation at any point situated at a given distance from 

 the axis. 



11. Accordingly, reverting to the expression of -fr in art. 7, 

 we shall now suppose that 



71% t 71V '-A./* 



/being equal to 1— er 2 . Also since the numerical value of k 



given in art. 7 was derived from the equation # 4 = -2 — r, and 



2k 2/c 5 



A was substituted for -^-g — rr, it follows that A = -~-. Hence, 



putting cr for p — 1, w for -yL, co for -—-, and deducing from 



N2 



