Mr EARNSHAW, ON FLUID MOTION. 211 



the medium, unchanged as to their nature only, but not as to their 

 magnitude. Thus, if the original disturbance produced cylindrical wave- 

 surfaces of concentric circular bases which at a given instant had certain 

 magnitudes, the wave-surfaces would continue cylindrical throughout 

 the motion, but the common centre of the bases, or the common axes 

 of the cylinders would be transmitted in the direction of x with the 

 velocity r/,7', and in the direction of y with the velocity f/,r, and the 

 radii of the cylinders would constantly undergo variation of mao'ni- 

 tude. ■ "^ 



10. These are the principal results of a general character which I 

 have been able to obtain. There is yet to be considered a certain 

 integral of tlie equation of continuity ; namely, 



= Clog.- + C', 



where r^ = {x - af + (y - /3)= ; which has hitherto formed the basis of 

 investigations in this part of fluid motion. 



Now in the general integral for a single disturbance, namely, 

 = F{f{x-a) +g{!/-fi), f, ff] + F, \/{x-a) -g{>J-li),f,g}, 

 let the forms of F and F, be assumed to be logarithmic, then 



C r 



•^ = 2 log {/(^ -«) ^giy-m + 2 log {f{x-a) -g{>j-li)\ 



= ^-^Og{f^x- af-g^{y - (iY\ 



= 2-iog(/^0; ■■r=-g' 



= C\ogr + Clog/ 

 = Clog/- -h C 



From this investigation it appears, that this integral is not equi- 

 valent to the general integral found by the usual process of integration, 

 but is in reality a very particular integral, applicable only to those 

 disturbances in which the sequence of the wave-surfaces can be ex- 

 pressed by logarithms. 



Vol. VI. Part II. Ee 



