424 Mr. "Tovey's Researches in the 



become, for these values, so large as to make the parts of the 

 sums S which contain them so much greater than the other 

 parts, that the latter may be neglected. Accordingly we will 

 assume this to be the case; and then the first of the equations 

 (3.) becomes 



This equation, being symmetrical with respect to x and y, 

 gives for v^ the same value whether x coincides with Ox, or 

 with O y, (fig. '2,). We shall therefore assume that v^ is sen- 

 sibly the same for all values of d. And then if we put 



we have v, = c. (5.) 



(10.) Now conceive a number of plane waves, pei'pendi- 

 cular to the plane of x^Oy^, (fig. 2,) all of which, at the 

 commencement of the time t, pass through the centre O ; and, 

 since v, is the same for all values of fl, conceive the velocities 

 of these waves to be all equal; then their distances from the 

 centre O will constantly be equal, and the curve, in the plane 

 of x^Oyj, touched by all of them at any instant will be a 

 circle. 



(11.) If the system of coordinate planes be turned on the 

 axis of x^ the circle and ellipse (art. 10 and 8) will describe 

 a sphere and spheroid.. And since this turning of the coor- 

 dinates will not, by the supposition (art. 4), sensibly affect the 

 values of the sums, and consequently not alter those of v, and 

 Vj^, it follows that the agitation at the centre O will in ge- 

 neral produce two sets of waves; of which one set will be 

 spheroidal, and the other spherical : the vibrations in the 

 spheroidal waves being perpendicular to the axis of x^ , and 

 the vibrations in the spherical waves perpendicular to those 

 in the spheroidal. 



(12.) From the supposed arrangement of the molecules 

 round the axis of x, it follows (art. 6 and 9,) that c := c,, 

 and consequently that when fl is zero we have u = v,, . Hence 

 by limiting our view to a spherical and spheroidal wave, both 

 of which emanate from the centre of agitation at the same 

 instant, we perceive that they will constantly coincide along 

 the axis of x . And when 6 is a right angle we have Vf, — c„, 

 which shows that the spherical wave will include, or be in- 

 cluded by, the spheroidal wave, accordingly as c is greater or 

 less than e^. 



By referring to Professor Airy's Mathematical Tracts, 

 p. 34,G — 350, it will be seen that the results obtained in this 

 and the preceding article are sufficient to explain the prin- 



