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EXAMINATION OF THE VALIDITY OF AN APPEOXIMATE 

 SOLUTION OF A CEETAIN VELOCITY EQUATION. 



By A. Brown, M.A., B.Sc. 



(Eead November 28, 1907.) 



§1. This problem has been suggested by the important part 

 played by " initial conditions " m the solution of differential equa- 

 tions occurring in Dynamics. In certain cases the special character 

 of the " initial conditions " makes a simple treatment of the equa- 

 tion possible ; but sufficient attention has not been paid to the 

 question whether the simplification introduced is legitimate, or to 

 the investigation of the limits within which it holds. 



To make the matter clearer, consider the equation — 



with initial condition y = X when t = 0, X being supposed small. 



One method of obtaining an approximate solution is to start with — 



dt~ y 



and find the solution of this equation. The result is Xe~ f . 



It is then assumed that if the solution thus obtained is small for 

 all values of t, the result given is an approximate solution of the 

 original equation. 



The actual solution of the equation is X . e~\ -. - . ,,-. —^, and in 



order that this be expansible in powers of \ for all values of t it is 

 necessary that Xk<l. If Xk>l the solution of the simplified equa- 

 tion is not even approximately a solution of the original equation ; 

 and unless Xk is a small fraction the solution given will not be a 

 good approximation. The validity of the simplification depends on 

 the nature of the coefficients as well as on the smallness of X ; and 

 it seems important to investigate how small X must be for the 

 simplified form to be used, 



