212 Mb EARNSHAW, ON FLUID MOTION. 



II. By assigning other forms to F and F^, other, and some of 

 them very simple*, expressions for (p may be obtained, and as in 

 Art. 5 the character of the motion may be determined. But as re- 

 sults so obtained would only be of a partial application, it is unnecessary 

 to pursue the idea further. There is one form however which seems 

 deserving of some consideration, as it presents us with a species of 

 fluid motion entirely distinct from that denoted by the integral 



^ = C log r 4- C ; 



and of such a nature as was supposed by Euler to make udx + vdy not 

 a complete differential; it is the following, 



The equation of a wave-surface being in general (Art. 6) 0=constant, 

 will in the present case be reduced to 



y-^=J\t).{x -a), 



wliich is that of planes intersecting each other in a line, parallel to the 

 axis of %, which passes through the point (a, /3), which is moveable or 

 fixed according as a, /3 are or are not functions of t. And since the 

 motion of every particle is perpendicular to its wave-surface, the general 

 motion of the fluid at any instant will be in arcs of circles having 

 their centres in the line which passes through (a, /3) : and the velocity 

 of a particle being 



= V(,d,<t>f + (.dy<pf = ±^, 



in this case varies inversely as its distance from the centre. 



The law of velocity in this instance is therefore the same as in 

 the one last considered ; but while that velocity carried the particles 

 towards or from a fixed or moveable centre, it here carries them round 

 that centre in such a manner that all of them describe equal areas 

 in equal times about it. 



• A very simple one is = a . i' '■'""'•. cos i (^ -/3). 



