390 PROFESSOR CHALLIS, ON THE DIFFERENTIAL EQUATIONS 



respecting the velocity, the density, and the form of the surface of dis- 

 placement. This consideration will enable us to draw some inferences 

 in particular cases without having recourse to the general equation. 



For example, the forms of the surfaces of displacement being any 

 that we please, it may be assumed that a given surface of displacement, 

 that is, one with which the same fluid particles remain in contact at suc- 

 cessive instants, continues of a spherical form. Its differential equation 

 will then be, 



2x (x — a)dx + %y {y — (Z)dy + 2z (ss — 7) d% = ; 



and the equation itself, 



(x - af + (y - /3) 2 + (s- 7 ) 2 =iF; 



in which a, /3, 7, B, may either all be functions of the time, or part 

 constant and part functions of the time. Let, for instance, a, (B, 7 be 

 constant and B a function of the time. Then since 



<f> = (x- «)■+ (y - /3) 2 + (2 - 7 ) 2 - B>, 

 it will be seen that 



£-«<•-* £-«*.-» jf.rW 



because B is by hypothesis the radius of the spherical surface in suc- 

 cessive instants. Hence by substituting in the equation, 



the result is, 





2i?r+ 4iVi2 2 =0; whence iV=-^. 



Hence, as JV is not a function of the co-ordinates x, y, %, the differ- 

 ential udx + vdy + wdz is integrable of itself, which for this case it 

 plainly should be, the motion being directed to or from a fixed centre. 



