﻿48 Dr. J. Gr. Leathern : Effect of a Long 



experiment, neglect the last term and consider, in the first 

 instance, only the equation 



dx 2 dx 



That this step is justifiable cannot be guaranteed before- 

 hand ; it must be tested by the results obtained. Two in- 

 dependent particular solutions of the curtailed equations are 

 y = a, and y = b\ogx, where a and b are constants. The 

 latter of these expressions for y is seen to be such as to make 

 the neglected term very small in comparison with each of 

 the terms retained, it therefore justifies the method by which 

 it was obtained and is a genuine approximation to a solution ; 

 moreover it becomes very great as x decreases, so it must be 

 an approximate form of that solution which we have called 

 h(x). The solution y = a, on the other hand, does not make 

 the neglected term smaller than those retained ; nevertheless 

 it makes two of the terms zero and the third very small, so 

 it is clearly an approximation to a solution. To improve the 

 approximation we try y = a(l + ^x + fix 2 ) ; substituting this in 

 the complete equation, and neglecting powers of x higher 

 than the first, we get, 



(4 A 6 + 1)^ + X=0, 



which is satisfied by X=0, fi= — \. ISo we get an approxi- 

 mate solution which, being finite for # = 0, must be y(x), 

 namely, 



g(x) = a{l-ix' 2 ). 



Differentiating we get further 



g' (x) = — \ax, h'(x) = b/x. 



Case of x very great. 

 In seeking approximations to the solutions when x is very 

 great, we note that the middle term of the equation has now 

 the appearance of being relatively unimportant ; so we 

 neglect it provisionally, and consider only 



Two independent solutions of this are y = a'e~ tx and 

 y = b'e ix , where a' and V are constants ■ and i z =— 1; these 

 forms are found to make the neglected term of the differ- 

 ential equation small compared with each of those retained, 

 and so the method by which they were arrived at is justified. 



When x is complex one of these solutions is zero and the 



