118 Mr. K. Moon on the Theory of 



that under all circumstances when v = we have p = a?p i that 

 would only entitle us to assume that 



p = a 2 p + Y b {p,v), 



where F& is some unknown function of p and v which vanishes 

 when v = 0, the making F 6 = when v does not =0 being to 

 be regarded as a pure unqualified assumption. 



Returning from this digression — since p may be regarded as 

 a function of p and v, we shall have 



^ = ±dp + ±dv =n dp_^ Y ch 



dx dp dx dv dx dx dx ' 



where R and V are functions of p and v only ; and substituting 

 this value in (1), observing that 



dy n dy 



v= d J> P = V'dx 

 we get 



dhj V d*y <k 



dt* Vdxdt dx 



■ 2 d?y 

 dx* 



P) 



in which, by virtue of the above equivalents for p and v, V and 



R may be considered as functions of — and — only. 



In treating of the integration of this equation, I shall assume 

 that V and R are definite functions, i. e. that they contain neither 

 arbitrary constant nor arbitrary function. If, by reason of the 

 expression for p in terms of p and v (and therefore in terms of 



-~j -j-) involving arbitrary constants or arbitrary functions, such 



or the like constants or functions enter into the composition of 

 (2) when exhibited in its most general form, I shall assume that 

 definite values have been assigned to such constants and func- 

 tions ; so that it is only to the equation (2) when modified in the 

 manner just indicated that the remarks immediately following 

 must be conceived to apply. 



Thus much being premised, I shall simply assume that (2) is 

 integrable by Mongers method, and shall follow out some of the 

 consequences of that assumption. 



One consequence of that assumption will be that (2) is capable 

 of being derived from a single partial differential equation of the 

 first order, of the form 



i^i^M^Wl 



(2 a) 



