Pressure in Fluids. 123 



of f a as those at which we have already arrived for the determi- 

 nation of/. 



Thus the occurrence of two of the four values of which /has 

 been shown to be susceptible is at once accounted for ; and if we 

 reflect that almost universally when an equation of the second 

 order is integrable by Mongers method, there are two equations 

 of the first order of the form (2 a) from each of which it is sepa- 

 rately derivable, there cannot, I think, be a doubt that the four 

 values above derived for/ must be attributable to the values of/ 

 and/ a in equation (12), and to the corresponding functions iu 

 the other first integral of (2). 



Acting from this clue, we shall be justified in assuming, and 

 we shall find it to be the fact, that when the equations of condi- 

 tion (LO) are satisfied, (2) will be susceptible of the two following 

 integrals, from each of which separately it is capable of being- 

 derived, viz. 





(13) 



If we now recur to the equations of condition (10) or (10«), 

 or, as they may be written, 



= P 9 ^-2R^ 



£ % 



dp dv 



where 



P,=V + */V 2 + 4V, P 2 =V- \/V 2 + 4V, 



the form of the equations leads us to conclude that, if neither of 

 the quantities P 2 P 2 vanishes (as for instance PJ so as to render 



—r^ = — -J =0, we shall have 

 dp dv 



p,=p. 



<2> 



which implies that we have v'V 2 + 4R/d 2 = 0, — a conclusion which 

 I have aleady pointed out as one to be rejected as deficient in 

 point of generality*. 



* This may be seen more distinctly as follows. In the case supposed, 

 (13) become 



V+l*-.aj'}'' *>"*!•- a'} ' 



whence we get coi= funct. w 2 ; and therefore we must have p = funct. of v, 

 since co, co 2 involve p and v only. Now not only is this conclusion, viz. that 

 p = funct. v, clearly defective in point of generality, but, if it were true, we 



