216 Mr EARNSHAW, ON FLUID MOTION. 



15. If we take the equation 



= df(j) + a' {d/<l> + dy^tf) + r//0), 

 its integral is 



<p = F, {/{x -a)+g(y-(i) + h{x-y)+ at,/, g, h} 

 + F, \f{x -a)+g(,j~ti)+h{z-y)- at, f, g, h} 

 + F, { -fix - „) + ^(y - /3) + k (%-y) + at, /, g, />} 

 + F. { -f(x -a) + g(rj -l3)+h{z-y)- at, f, g, h\ 

 + kc. ... 



subject to the condition 



fi +g'- + h- + 1=0. 



Each set of values of f, g, h Avill furnish eight arbitrary functions 

 in the value of cp. As there are no arbitrary functions of t to be 

 added to complete the integral, as was necessary in incompressible 

 fluids, t enters only in the form a;bove exhibited ; and it is evident 

 that the integral is of such a nature as to render it impossible to make 

 / disap'pear by changing the origin (as in Art. 9, II.) : wherefore the 

 reasoning of (III. Art. 9) can be applied here; from which we infer, 

 that the extent and intensity of the disturbance are continually chang- 

 ing, inasmuch as the equation of the wave-surface does not change its 

 form, but only the magnitude of its parameters which are functions 

 of /. As a wave-surface expands, a point which has a certain 

 relation to it remains fixed in the medium : so that the expansion 

 may be said to take place about this point. 



16. The following is also a solution of the differential equation 

 of last Art. 



r(p = F{r + at V^^) +f{r - at \/^^) ; 



;• being = \/{x - af + (y - (if + (« - 7)'- 



And by assigning particular forms to F and f, we shall obtain cases 

 of possible motion ad libitutn. 



