444 Mr STOKES, ON THE STEADY MOTION 



* ^ifo?) + \-dfi J + * ^Uf + Wi = °' 



f ([/,) rf^ + rfy _ 



^'(t/-,) /^'x ffiy 



+ 



Now, if the motion be possible, the second term of this equation 

 must be a function of £7, ; #, # and f7, being connected by the equation 

 «/*(#> y) = E7i. Consequently, if by means of this latter equation we 

 eliminate x or y from the second term of (6), the other must disappear. 

 If it does not, the motion is impossible; if it does, the integration of 

 equation (6), in which the variables are separated, will give <p(U t ) under 

 the form 



cj>(U i ) = AF(U 1 )+ B, 



A and JB being the arbitrary constants. The values of u and v will 

 immediately be got by differentiation, and then p will be known. 

 Nothing will be left arbitrary but a constant multiplying the values 

 of u and v, and another added to the value of p. 



I shall mention a few examples. Let U = ar% cos \ 8. In this case 

 the lines of motion are similar parabolas about the same focus. The 

 velocity at any point varies inversely as the square root of the distance 

 from the focus. 



Again, let U = axy. In this case the lines of motion are rectan- 

 gular hyperbolas about the same asymptotes. Also, 



dU , dU 



u = -j— = ax, and v = *- = — ay. 



dy dx * 



In this case therefore the velocity varies as the distance from the centre, 

 and the particles in a section parallel to either of the axes remain in 

 a section parallel to that axis. 



I shall now consider the general case, where udx + vdy need not 

 be an exact differential. 



