446 Mr. STOKES, ON THE STEADY MOTION 



and therefore, 



5- '-*{(£)'♦ m*M*%) (£ a ' + £*)- 



It will be observed that ^-j- + -5-5- = x (U), is a first integral of (10). 



Consequently this latter term, which is the value of C in (1), comes out 

 a function of the parameter of a line of motion as it should. 



We may employ equation (10), precisely as before, to enquire whether 

 a proposed system of lines can, under any circumstances, be a system of 

 lines of motion. Let fix, y) = U l = C, be the equation to the system ; 

 then, putting as before, U = <p ( ?/,), we get 



+*fwtb,( du * d dU > d \l( du *Y-L_ ( dU >V\ 



r ^Uy" Tx~ ~dx~ dy)\\dx) + \dj) j 



+ * {U ^ \-dJ dx ~ ~dx~ dy-) VdlF + ~dtf) = ° ; 

 or, Ptf'iUJ + Q0'(E7,) = 0, suppose. 



Hence, as before, if we express y in terms of x and U lt from the 



equation f(x, y) = U lt and substitute that value in -5, the result must 



not contain x. If it does, the proposed system of lines cannot be a 

 system of lines of motion ; if not, the integration of the above equation 

 will give tyiUy), under the form <£(?/,) = AFiUi) + B, and we can 

 immediately get the values of u, v and p, with the same arbitrary con- 

 stants as in the previous case. 



One case in which the motion is possible is where the lines of motion 

 are a system of similar ellipses or hyperbolas about the same centre, 

 or a system of equal parabolas having the same axis. In the case of the 

 ellipse, the particles in a radius vector at any time remain in a radius 



