of the Motion of Fluids, See. 385 



origin of co-ordinates and is equal to -^. The integral of 'Jf = o, 



which is, (p = fit) + — — , determines the velocity ^ to be 



— F (t) 



— ^3 — , and thus gives the law according to which it varies at 



different distances from the point to or from which the motion 

 tends. This law may be verified by conceiving a small spherical 

 ball, capable of expansion, to be placed concentric with a sphe- 

 rical fluid mass, inclosed in an envelope also capable of expansion. 

 By the expansion of the ball, the particles will be moved through 

 spaces which vary inversely as the squares of the distances from 

 the centre. The supposition that ^ is a function of r and /, does 

 not necessarily restrict the application of the preceding integral 

 to a particular case; for the law of motion it discloses must 

 obtain wherever the parts of the fluid act on each other. More- 

 over, if the equation, d<p = udx+vdi/ + wdz, has the meaning above 

 assigned to it, at one point, it must have the same at every 

 point; and if it be applicable to every point at one instant, it 

 will, as Lagrange has proved, be applicable to every point at 

 every instant. We may consider, therefore, the preceding integral 

 to have been obtained on the supposition that the origin and 

 direction of co-ordinates were arbitrary, and consequently to be 

 applicable to every point in motion*. It is necessary to suppose 



* Similar reasoning is applicable to that integral of 



which is obtained by supposing (^ to be a function of r and t. I have not clearly stated 

 in Art. 14, of my Paper on the Small Vibrations of Elastic Fluids, {Cam. Phil. Trans. Vol. III. 

 Part. 1.) upon what principle this integral may be considered general. It is general, as 

 regarding the mode of action of the parts of the fluid on each other. M. Poisson's integral 

 of the same equation {Mem. Acad. Sciev. 1818.) is general in a diflcrcnt sense; — in regard 

 to its application to any proposed instance. 



3 d2 



