226 Mr EARNSHAW, ON FLUID MOTION. 



f being an arbitrary constant. This quantity gives rise to a difficulty. 

 For though the proposed problem is sufficiently specific, yet it affords 

 no data for the determination of f, and consequently the curve sought 

 remains as vui determined, and the problem apparently as unsolved, as 

 ever; all that Ave can gather from the above integral being, that a 

 straight line whose position depends on the value of f will fulfil the 

 proposed conditions. 



It ma}' be said, however, that any value given at pleasure \o f will 

 determine a line answering the conditions of the question ; but it is 

 clear that a line so found can only be considered as a partial solution ; 

 inasmuch as the fixing upon a particular value of f tacitly implies the 

 possession of data enabling us to decide upon that value in preference 

 to all others. Now such a decision cannot be received, unless the data 

 which led to it are supplied by the conditions of the proposed problem ; 

 and, as we have seen, no such data exist ; consequently no particular 

 value of f can be received. Thus it appears, that though the integral 



y =fx ± s/TTpTb"-* 



furnishes partial solutions of the proposed problem without number, 

 it does not present us with the required curve. The only resource 

 left is, to employ equally all possible values of f from — oo to + x ; 

 all having an equal claim to have weight in the general interpretation 

 of the above integral. This will present us with an infinite number 

 of straight lines, not drawn at random, but according to a law expressed 



by 



y =fx + ^/arf + V; 



and by the intersections of consecutive lines forming a curve to which 

 they all have an equal relation, being tangents, which is allowed to be 

 the curve required. 



♦■ Since / admits the sign - as well as + , we have, by taking all the variations of 

 sign, four straight lines; one in each portion of space comprehended between the co- 

 ordinate axes. See Arts. 18, 21. 



