OF THE MOTION OF FLUIDS. 185 



But as a, a, a", are the angles which Ox makes with three rectangular 



axes, 



cos" a + cos" a + COS" a" = 1, 



so cos-/3 + cos^/3' + cos^/3" = l, 



and cos'^7 + cos'^7' + cos^7" = l. 



Therefore by adding the equations (1), (2), (3), 



du dv dw _ 

 7lX^ dY^dZ~ 



11. The general conclusion from all that precedes is, that the law 

 of the variation of the velocity from any point to another indefinitely 

 near in the direction of the motion, at a given instant, may be expressed 



C 



by -^ — J-, the quantities C, r, and I, being such as we have stated 



C 



in Art. 9- If 1=0, we have- as a particular case, V=-^. In my 



former paper on the motion of fluids, I assumed, as it now appears, 



C 



incorrectly, that — represents the general law of the variation of the 



velocity. None, however, of the results in that paper are affected by 

 the assumption. For instance, the expression for 



as it only requires that (p should be a function of r and /, will remain 

 the same. This expression may also be briefly obtained thus. We 



have seen that -~- = V. Now as r is ultimately in tlie direction in 



which the velocity V takes place, if a line commencing at a given 

 point be drawn constantly in the direction of the motion at a given 

 instant of the points through which it passes, dr may be considered the 

 increment of this line. Hence if we call its length s reckoned from 



the fixed point, -j^ = -^ = F. Then integrating, c}> = jVds -^/{t); 



