0}i some Equations in Partial Differentials. 107 



le volume total en parties de grandeur finie, mais insensible, 

 dont chacune renferme neanmoins un nombre extremement 

 grand de molecules I'equation [of continuity] exprime reelle- 

 ment que chaciine de ces parties renferme toujours les memes mo- 

 lecules, et que par consequent sa masse est invariable." 



In all cases where new forces are being continually intro- 

 duced, as in the flowing of rivers or of water through pipes of 

 variable form, it is impossible to calculate the motion, and at 

 the same time the equation of continuity is no longer appli- 

 cable. But, on the other hand, there are cases in which this 

 equation holds, where nevertheless the other three equations 

 of motion do not hold, and in fact where there may actually 

 occur a split or division in the fluid. As an instance may be 

 mentioned the flowing of a stream of water over the edge of a 

 perpendicular precipice, where gravity suddenly comes mto 

 action. In this case the number of particles may remain the 

 same in an element which may be conceived to consist of par- 

 ticles that have fallen over as well as of those that remain. But 

 the equation of motion would require the new force of gravity 

 to be taken into account. 



XXIII. On tlie Integration of some Equations in Partial Dif- 

 ferentials. By the Rev. Brice Bronwin*. 



IN the present paper I shall extend the mode of integrating 

 certain differential equations, which I gave in this Journal 

 in December last, to the integration of similar equations in 



partial differentials. Let D stand for -p, D' for -r— ; and as 



a first example, let 



where p, as throughout the paper, is a positive integer. Make 



then 4^ + F.)+2;;^=0. 



This is the first example in the paper above referred to, 

 where by making s=(L)^ + ^^)'' u, it is reduced to 

 d'^u , c 



or, restoring the suppressed factor, to 



• Communicated by the Author. 

 12 



