OF THE MOTION OF FLUIDS. 181 



directions of the normals at right angles. The motion, being in the 

 normals, will be directed to the focal lines. If we describe another 

 surface indefinitely near the first, and cutting in like manner the direc- 

 tions of the motion at right angles, all the points of any fluid element 

 intercepted between two opposite elements of the surfaces, will at a 

 given instant ultimately have their motion directed to the same focal 

 lines : but this cannot be said in general of more than an elementary 

 portion. If we suppose the form of the superficial element to be a 

 rectangle, the normals through all the points of its sides, will inclose 

 a wedge-shaped mass, the transverse section of which at any point, it 

 is easy to shew, will vary as the product of the distances of that point 

 from the focal lines. Hence the velocity in passing at a given instant 

 from the first to the second of the surfaces above-mentioned wiU vary 

 inversely as this product. Let therefore r and r + l he the distances 

 of the point whose velocity is V, from the focal lines to which the 



C 



motion is directed. Then V= . j-, in which expression C, /, and 



the positions of the focal lines are constant at a given instant, when 

 r varies through a space which may either be finite or indefinitely small. 

 Let a, /3, 7, be the co-ordinates of the middle of that focal line which 

 is distant by r from the point in question. The velocity (m) in x will 



then be V. ; the velocity {v) in y, V. ; and the velocity 



{w\ in », V. ^. Hence 



' r 



udx + vdy + wd%= Vi — ~dx + ^ , dy H -d%\ . 



Now since r- = {x — af + {y — fif-\-{%~yY, if we make r vary with 

 X, y, and %, while a, )3, 7, remain constant according to what has just 

 been said, we shall have rdr — {x — a)dx + {y-fi)dy + {% — y)dti. Hence 

 tfdx + vdy + wdz=F^dr; and as F" is a function of r and /, the right 

 side of the equation is a complete differential of a function of 

 X, y, %, and t, with respect to the three first variables, t being con- 

 stant. Therefore also the left side is the same. Let the function be <p. 

 Vol. V. Part II. A a 



