for Differential Equations of Mathematical Physics. 593 



It has become apparent that for all practical purposes the 

 series used in the foregoing analysis are rapidly convergent. 

 In point of fact the rapidity of convergence can easily 

 be investigated, for each one of the functions f^x) . . . f(x) 

 is absolutely convergent. 



Let fi x and R x be the greatest and least values of p and R 

 over the range of x considered ; then, for example, 



I h W I < i + ^r 1 r^ I %dx K d * \ X(l * + • • • ; 



*m Jo Jo Jo Jo 



i.e., ^ l4 _^'V_i J^rf 



^ ± + 4! Ri 8! IV" 

 Hence 



nth term d^affjbi 



X 



n-ith term Ri (4n-4)(4w-5)(4n-6)(4n-7)' 



showing that the series converges with extreme rapidity. 



Part II. 



§ 1. General. — The equations of the flow of an incom- 

 pressible viscous fluid stand almost unique in mathematical 

 physics in virtue of the fact that no single unrestricted and 

 complete solution has yet been obtained in any problem. 

 Failing a complete mathematical breakdown of the equations, 

 attempts have been made to derive information regarding 

 fluid-motion phenomena by the elimination on the one hand 

 of one of the most vital characteristics of the fluid, viscosity, 

 and on the other by such limitations on the motions as would 

 involve no effect due to the inertia of the fluid. From the 

 physical standpoint, however, based on the accumulated ex- 

 perimental experience of these questions acquired in hydro- 

 and aerodynamical research, one is inevitably driven to the 

 conclusion that both these factors are of prime importance 

 and must play a prominent part in any explanation of the 

 characteristics of fluid motion. 



Certain outstanding experimental facts found by previous 

 workers may here be mentioned, as they bear critically on 

 the problem at issue. Osborne Reynolds * discovered the 

 existence of a critical state of flow originating during flow 

 of water in pipes above a certain value of vl/v, where c is 

 the mean velocity in the pipe, / the diameter, and v the 

 viscosity of water. Further investigations on this point by 



* Phil. Trans. 1883, p. 935. 



