V- 



328 Dr. 0. Burton on Plane and Spherical 



part of it will be instantaneously moving forward with (very 

 approximately) the velocity 



•*? + 



dp 



determined by the corresponding values of p and u ; so that 

 M will be gaining on N at the rate 



Au+ i\/¥/£ Au ^ 



approximately. We may admit then that the rate at which 

 M gains on S" is 



never < BAw, 



where B is a constant suitably chosen. 



Again, if A u is the difference between the air- velocities at 

 M and N at the time £ = 0, and r is the corresponding co- 

 ordinate of M, we may admit that 



Au is never < — A Q u, 



r + at 



where A is a constant not very different from unity. Thus 

 M gains on N at a rate which is 



never < AB r ° & Q u ; 

 r + at 



and between the times t=0 and t=t x the distance gained by 

 M relatively to N will be 



at least ABAo^f^^L, 

 Jo r o + at 



i.e. at least ABA w^log^±f^. . . . (14) 



If B is finite and positive this expression increases indefinitely 

 with the time, so long as the laws of continuous motion hold 

 good. If A r was the distance between M and N at time 

 t=0, the time required for M to overtake* N will be not 

 greater than the value of t x given by 



-A r=ABA iAog^±^; 

 a ° r 



or, when M and N are taken indefinitely close together at 

 starting, by 



log?^=^ JABf-^U; 



i.e., we have t x = !^f D * ^ AB ( - S04 _ i \ u % (15) 

 * Cf. Lord Rayleigh, 'Theory of Sound/ vol. ii. p. 36. 



