392 PROFESSOR CHALLIS, ON THE DIFFERENTIAL EQUATIONS 



At the same time equation (14) to the same degree of approximation 

 becomes, 



And if P = 1 + % P = a\ Nap. log (1 + 8) = aU nearly. Also 



dV - „ f(r-at) fir -at) 

 at r r 



Hence / -rrdr = — « - — -. 



J at r 



and F.(t) + a*$ = a/'^ ~ at) . 



When the motion is directed to a fixed centre, these results apply 

 to the whole of the fluid in motion, and at any time. When the centre 

 is moving, r is in general a function of the co-ordinates of the point con- 

 sidered, and of the time, and the same results are applicable to all 

 the points for which this function can be assigned. For instance, in 

 the example considered above, in which the surface of displacement is 

 that of a sphere of given radius, the centre of which moves in a straight 

 line, r is at all times constant for all points of this surface. We have, 

 therefore, from what is shewn above, 



%- y ^y _ f'(R - at) f(R - at) 

 R dt ~ R R> 



an equation which is true whatever be t. Let -y- , the velocity of the 



centre of the surface, be mtp(t), and let — ^ = cosfl, being the angle 



which the radius of a point whose co-ordinate is % makes with the line 



of motion. Also let f(R- at) = f. Then f(R-at) = --<¥. Conse- 



' a at 



quently for determining f we have the differential equation, 



4- +^ + maRcos9(p(t) = 0. 



