Modified Theory of Gravitation. 81 



which expression must therefore be equated to v 2 $ das dy dz, 

 6 being the velocity-potential o£ the secondary motion now 

 considered, while y 2 = 373* 2 + V[b y 2 + dW- 



With due regard to boundary conditions, this leads to 



*=lW^ d *' dy ' dz '-> ■ ■ ■ (n) 



<j' being the density of atomic matter at any point (a?', y\ z'\ 

 while r 2 =(x-x') 2 + (ij-y') 2 + (z-z') 2 . " The integration 

 must be conducted through a sufficiently extended region to 

 include all pulsatory centres which contribute appreciably 

 to the value of </> at (#, y, z). In accordance with the as- 

 sumption mentioned in § 15 above, the factor ~dp/'dt is placed 

 outside the sign of integration ; while sources and sinks other 

 than those corresponding to the expression (10) are assumed 

 to be absent from the secondary motion. 



17. Treating the primary wave-motion more generally than 

 hitherto, (4) may be taken as expressing a single typical 

 constituent, no restriction being imposed on the direction of 

 propagation of the waves, or on their periods or phases, 

 provided that all effective wave-lengths are supposed suffi- 

 ciently great, and all limited sources of the primary 

 disturbance sufficiently remote. The form of our equations 

 of course involves the implicit assumption that, in the 

 primary wave-motion, all relations are linear. It is now 

 similarly assumed that, in dealing with the secondary dis- 

 turbance expressed by (11), non-linear terms may be omitted 

 from the equations, the principle of superposition being 

 accordingly applicable to the primary and secondary motions. 

 Thus, to our order of approximation, we may write 



and 



2>=i* + P + «- (12) 



= _ _AV J + f unCt ^ . 



(13) 



where <f>, tv are respectively the velocity-potential and pressure- 

 increment corresponding to the secondary motion, -v/r, p being 

 the velocity-potential and pressure-increment corresponding 

 to the primary disturbance. Hence 



~dx\-dt ^'dtj- p-dx" AcU + W 5 ' ^ J 

 Phil. Mag. S. 6. Vol. 17. No. 97. Jan, 1909. G 



