OF THE MOTION OF FLUIDS. 187 



in the direction of the motion at the points through which they pass, 

 these lines must be rectiUnear, because there is no curvilinear motion 

 at the boundaries of the fluid, and therefore no cause to impress a 

 curvilinear motion on the parts interior. The straight lines commencing 

 at P and R will intersect ah and ae at p and r, points equidistant 

 from a, and those commencing at Q and S will intersect the same 

 lines at q and s also equidistant from a. Now from the law of 

 the variation of the velocity above found, at every point of the cunei- 

 form element Ps, the velocity will be inversely proportional to its 

 transverse section. Let therefore V =^ the vertical velocity with which 

 (lb is made to descend, and v the vertical velocity with which the 

 surface DB descends. Let AB = a, AQ = x, PQ = X, ah = h, aq = x', 

 pq = 'S.', and the angle BAE = e. Then the element PQSR^^xeX, 

 and pqsr = x'e\'. These elements are proportional to the transverse 

 sections at P and p ; and the vertical velocities V, v, are to each 



other as the velocities at p and P in Pp. Hence — = -; — , = -V-, • 



-' ■* V X e\ xX 



F • Wence ^, - ^ 



because the motion is along the slant surface. Therefore in this case 



X a 



r-, =" T. Suppose X to be given, and let Xi be the consequent value 



X o 



of x'. Then — = -r, and -. = y . If now x be taken = « — X, from 



X, o b-Xi o 



what has been just shewn, x' will = 6 — X, . Hence 4 — = j^, and 



\0 — Xi) X2 o 



consequently — = t- Therefore X2 = Xi; and so on. From this it 



X2 o 



follows that if AB and ab be divided into the same number of 

 indefinitely small equal parts, the straight lines joining the corresponding 

 points of division will give the directions of the motion, which is 

 consequently every where directed to the vertex of the cone. Hence the 



velocity af^ any point W whose distance Cp W from C is p, varies as — . 



P' 

 Let CA =h, Ca = k, z AC W= 9 ; then the velocity at p=V sec 9, and 



the velocity at W= V sec 9 . ^ — ; this resolved in the vertical 



But — also = jj, . Hence ^j^, = 71 • If we take x = a, x' must = h. 



