OF THE MOTION OF FLUIDS. 177 



supposed to vary in any manner along this line. The foregoing- 

 reasoning only proves that in passing at a given instant from one 

 point to another indefinitely near along the line, these quantities may 

 be considered constant. 



4. The nature of the integral we have been discussing will perhaps 

 be understood by comparing it to the general integral of a common 

 differential equation, which has a particular solution. The latter, we 

 know, is that which gives the answer to a proposed problem, and the 

 general integral is used (though not necessarily) to obtain this solution. 

 So, I conceive, the integral above is useful for arriving at the particular 

 functions of x, y, and t, which give the velocity and direction of the 

 velocity at any point and instant in any proposed question. The 



integral of -—^ + -r^ = , which M.M. Poisson and Cauchy have 



obtained for the solution of the problem of waves, may be called the 

 particular solution of the equation, for that particular problem ; and I 

 think it probable that the same might have been obtained by employing 

 what I would call the general integral, though I am not prepared to 

 state exactly the process. 



5. The following considerations are added in confirmation of the 

 foregoing reasoning. In whatever manner the fluid is put in motion, 

 we may conceive a line, commencing at any point, to be continually 

 drawn in a direction perpendicular to the directions of the motions at 

 a given instant of the particles through which it passes. This line 

 may be of any arbitrary and irregular shape, not defined by a single 

 equation between x and y. But it must be composed of parts either 

 finite or indefinitely small, which obey the law of continuity. Con- 

 sequently the motion, being at all the points of the line in the directions 

 of the normals, must tend to or from the centres of curvature, and 

 vary, in at least elementary portions of the fluid, inversely as the 

 distances from those centres. An unlimited number of such lines 

 may be drawn through the whole extent of the fluid mass in 

 motion. 



