OF THE MOTION OF FLUIDS. 183 



through the same point p, a; = «'x = a« + /+r, and therefore ^ = -7 . 



And y = hz = h'% ^i therefore z = -rm — tx • Hence -; = —rrr, — 7t> 



'' a' a{b'-h) a -a a{b-b) 



which gives h' = — ; j^. From p draw ps perpendicular on Ox, 



and let P.? = 5. Then ^ = x-{r + l). Bnt x = a'z = t!^±Il, Therefore 



' a —a 



^ = — 7 '- . Hence it will be found that a' = — — » , and 



a —a 6 



V = — J — -. This latter quantity, if we neglect powers of S above 



the first, is equal to (l H — rj A. Therefore by substitution 



„ d r \ r{l + r)J 



cos / Jr«o = — -. . , , ,. 



a'(^r + l)+(l+d' + b'—]s 

 = (neglecting ^, &c.) V ^ / 



Here / „ === is the cosine of the angle pPs. Hence if ^ = the 

 V 1 + a^ + o^ 



velocity at P in Ox, and w the part resolved in the direction parallel 



Va 



to AZ, w — — / 2~ ^i • ^^^ ^ ~ ^^ resolved portion of the velocity 



at p in the same direction. Now the velocity at p is ultimately the 



same as that at s, and is therefore equal to V . -, A^ \ — r- , 



according to the law of variation from P to s determined . by the 



AA2 



