180 Mr CHALLIS's RESEARCHES IN THE THEORY 



in less time than those more remote. This is particularly observable 

 in two of the floating particles which are near each other, and at nearly 

 equal distances from the centre. That which is less distant overtakes 

 the other, as it ought to do, supposing it to describe a less circle with 

 equal velocity. At the centre a kind of eddy is formed, the more 

 observable as the motion at every point of the surface is more nearly 

 in concentric circles. When the revolving motion takes place in a 

 conical tunnel from which the water is issuing, the appearance at the 

 axis is very remarkable, a hollow space like a sack, being formed a 

 considerable way down the axis. What has been here said may serve 

 to explain in some measure the manner in which eddies in any case 

 are produced. 



SECTION II. 



On the Integration of the Equation -^ + -r^ + -r^ = 0. 



9. M. Poisson has expressed the general integral of this equation 

 by means of definite integrals ; {Memoires de rAcademie des Sciences, 

 Ann. 1818), and this, I believe, admits of a discussion similar to that 



applied above (Art. 3.) to the integral of -^ + -~ = 0. But perhaps 



the following reasoning, analogous to what was indicated in Art. 5., 

 may be considered sufficient. In whatever manner the fluid is put in 

 motion, we may conceive a surface to be described, which shall be 

 every where perpendicular to the directions of the motions at a given 

 instant of the particles through which it passes. This surface may be 

 of an arbitrary and irregular shape, not necessarily defined by a single 

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

 finite or indefinitely small, which are continuous, and consequently have 

 radii of curvature subject to the same conditions as those of regular 

 curve surfaces. Hence the normals to all the points of any element 

 of the surface will pass through two focal lines, situated at the centres 

 and in the planes of greatest and least curvature, and cutting the 



