334 Mr. J. Buchanan on the Theory of 



With these assumed values for F (a) we get by integration 

 of (6) 



V4H 



(9 /»V4(H-A) x 

 ^J ^) (?> 



Verify (7) by observing that, when #=0, I = «H if H has 

 any value between the limits and h ; and I = c if H lies 

 between the limits h and oo . 



(7) also satisfies the differential equation (3). It is also 

 evident that in (7) 



when H=0, 1=0 ; and -^ =0 for all values of x 

 except #=0, when -^ —a. 



dR 

 ForH = co, I = c, and :7Lf = 0« 



For ,h'=gc> , 1=0. 



Hence the graphs which represent (7) fulfil the same 

 terminal conditions as the experimental curves, when H is 

 increased continuously. 



It will be observed that for large values of H the right- 

 hand member of (7) is sensibly equal in value to the last 

 term, viz., 



--K 1 - ^Jo e -4 • • (8) 



Partly in order to get an idea of the march of I, as H and 

 x are altered, for other than moderate values of H, I have 

 drawn some graphs of (8) . These are shown in fig. 3. 

 They indicate plainly that as w increases the curves get 

 flatter, but they all have as asymptote u = c : for convenience 

 I have taken c = 1 . 



The numerical calculation of the points on the graphs of 

 the complete equation (7) is very laborious, so that I have 

 been obliged to content myself for the present with the 

 graphs shown by full lines in iig. 1. H is taken as the 



