Mr EARNSHAW, ON FLUID MOTION. 207 



6. It does not appear possible to carry these investigations much 

 farther in a perfectly general form ; it will be necessary therefore to 

 introduce the hypothesis of the expression udx + vcly + icd% being 

 integrable per se. Denote its integral by ^, then 



= constant =f{f). 

 will be the equation of a wave-surface. 



The effect of this hypothesis will be, to exclude from our re- 

 searches many cases of motion in wave-surfaces, and all motion in 

 wave-lines. 



FLUID MOTION OF TWO DIMENSIONS. 



7. I have preferred commencing my investigations with this simple 

 case because the results more frequently admit of perfect investigation, 

 and are more easily and briefly expressed in Avords than in the case of 

 three dimensions. 



The equation of continuity now to be considered is 



d:<^ + rf/0 = 0, 

 and its integral is 



^ = F \f{x-a)\g(:y-Q),f,g,t\ ■ 

 + F,{f{x-a)-g(y-fi),f,g,t\; 



subject to the following condition between the arbitrary constants y and ^^f, 



r +g"- = o. 



In this integral the forms of the functions F and F^ are perfectly 

 arbitrary, to be adapted in any example to express the law of sequence 

 (as to space) of coexistent wave surfaces, according to the nature of 

 the original disturbance. The arbitrariness of these functions shews 

 that the fluid can transmit a disturbance of any kind which does not 

 violate the continuity of the fluid, a, /3 are arbitrary constants enabling 

 us to fix the origin of co-ordinates in the most convenient position : 

 they may besides contain functions of /, which depend upon the nature 

 of the original agitation. The functions of ^ g, t, which enter under 



