364 PROFESSOR CHALLIS, ON A THEORY OF LUMINOUS RAYS 



being p and density p at the time t at any point whose co-ordinates are <v, y, x, it will be 

 assumed that p = a'p, a? being a certain constant. 



In a former communication which I made to this Society, I gave the proof of a new fun- 

 damental equation in Hydrodynamics, by the combination of which with the ordinary equation 

 of continuity, an equation results which is indispensable in the present investigation. The process 

 for deducing this last equation is given in the Cambridge Philosophical Transactions, (Vol. vii. 

 Part III. pp. 38,6 and 38(3) : it is also obtained (p. 387) by independent elementary considerations. 

 Let V be the velocity and p the density at any time t, at a point where the principal radii 

 of curvature of the surface cutting the directions of motion at right angles are R and R\ and 

 let ds be the increment of a line coincident with the directions at the time t of the motions of 

 the particles through which it passes. Then the resulting equation I speak of is, 



dp d.pV ,11 l\ 



the variation with respect to space being from point to point along the line s. Now the new 

 fundamental equation above mentioned, combined with the two other fundamental equations, gives 

 the means of obtaining a resulting equation, in which the variables are x//, x, y, x and t, the 

 principal variable \|/ being such a function of the othei's that \^ = is the equation of a surface 

 normal to the directions of motion, in whatever way the motion of the fluid may have originated. 

 It follows that the function \|/, since it is given by a partial differential equation, contains arbitrary 

 functions of x, y, z and i, and that the normal surface is consequently arbitrary. The partial 

 differential equations applicable to the Undulatory Theory of Light are linear with constant 

 coefficients. For our present purpose, we have to enquire how far \|/ is arbitrary when the 

 equations are of this nature : whether, for instance, the normal surface must necessarily be either 

 a plane or a spherical surface. The general equation which gives v// by integration is too com- 

 plicated to be employed in this investigation. We may, however, dispense with the use of it 

 by combining equation (l) with the following general equation, which is obtained in p. 383 of 

 the communication already referred to : 



rdV V 

 o=Nap.logp + J—ds + - = Fit), (2), 



the variation with respect to space being, as before, from point to point of the line of motion. By 

 differentiating this equation with respect to s and t successively we get, 



a'dp dV ^dV . a'dp fd'V ^ dV „_^^ . 



— ^ + — +V — = 0, and c + / ^s + V — = F (t). 



pds dt ds pdt J df dt ^ ' 



Also equation (l) may be put under the form, 



pdt pds ds \R R'l 



Hence substituting for — — and — —- from the preceding equations, and differentiating with respect 

 pdt pds 



to s, the result is, 



d'V d'V d'V dV dV „dV /] 1\ , /i in 



for dR = dR = ds. Now putting --- for F, and integrating with respect to « after the substitu- 

 tion, it will be found that 



