APPLICABLE TO THE MOTION OF FLUIDS. 379 



We may therefore conclude that in the instances of' fluid motion for 

 which udx + vdy + wdz w an exact differential of a function of three 

 independent variables, the motion of every particle of the fluid is rectilinear. 



7. Hence in equations (9) and (10) the radii of curvature r, r, pass 

 through fixed points or fixed focal lines, and the line * coincides with 

 r. Hence changing ds into dr in equation (9), substituting <p' + fF(t)dt 

 for (p, and suppressing the accent, we obtain, 



dQ d$ dQ ( d >_d£\d*$_d°$_ d$ d^ d$tl in 



dt + dr' dr + \ dr*) dr* d? dr' drdt + dr\r + r) ~ " A > 



which equation is applicable to any instance whatever of rectilinear fluid 

 motion. 



8. I proceed now to the consideration of the more general case, 

 viz. that in which udx + vdy + wdz becomes integrable by being mul- 

 tiplied by a factor*. Let -**. be the factor. Then we may assume the 

 function <p to be such that, 



(dtp) = jydx + ~dy + ^dz, 

 so that we have, 



dx dy d% 



The introduction of this new quantity N makes an additional equation 

 necessary by which it may be determined. This may be investigated 

 as follows. By the reasoning of Art. 3, it appears that 



jjdx + -^dy + jfdz = -^dr; 



and if the variation be from one point to another of a surface of dis- 

 placement dr = 0. Hence the equation, 



jydx + -^dy + jjdz = 0, 



* Mr. Earnshaw has suggested the idea of multiplying by a factor, in the Paper on Fluid 

 Motion already referred to. 



