Mr EARNSHAW, ON FLUID MOTION. 217 



Thus, let F and f be logarithmic, as in Art. 10. 



.-. r<p = C log (;• + at V -I) + C log (/• - at V'^) 

 = Clog(/^ + d'f); 



' r 



This will denote spherical waves converging towards their common 

 centre with the velocity 



r 



17. The following is also a solution : 



.0 = Ctan-.Q; 



which denotes spherical waves diverging from a common centre with 

 a velocity «. 



18. I come now to by far the most interesting case; namely, 

 that wherein the velocities of the particles of the fluid are small in 

 comparison of a, and where 



cl(^ = udx + vdy + wd%. 

 The differential equation for this kind of motion is. 



and its integral is 



= ^1 {/(^- - «) + ^-(y - /3) + h (a - 7) + at, f, g, h} 

 + F, {f{x - a) +g(tj - fi)+k(s - y) - at, f, g, h\ 

 + &c. ... 



the form of the integral being precisely the same as in Art. 15, but 

 the equation of condition among the constants f, g, h i& here 



f +g' + h' = i. 



We observe that / g, h, may all be possible, and in what follows 

 they will be supposed possible quantities. This circumstance will alone 



