*52 Mr. STOKES, ON THE STEADY MOTION 



whence £_- 4( , +1 ^ + /(*_£) !«, 



Hence the quantity under the integral sign must be a function of 

 17. And in fact, we can easily shew by trial that 



l (d*U d*U l dU\ 



^Vd¥ + -d^-r^J =>HC7) 



is a first integral of (21). The last term of (22) is the value of the 

 constant in (1). 



By expanding U in a series ascending according to integral powers of 

 «, which may be done as long as the origin is arbitrary, it will be found 

 that the integral of (20) may be written under the form 



U = cos (yx)F(r) + sin (v*) V _, /(0» 

 where v 2 F (r) denotes U*- - - -~-J F(r), and v" F{r) denotes that the 

 operation -y-j -7- is repeated n times on F(r). 



We may employ equations (21) or (20) just as before, to determine 

 whether the motion in a proposed system of lines is possible. If 

 F(r, ss) = C/i = C be the equation to the system, we must have, as before, 

 U = <p( Ui) ; whence we get, in the general case, 



*"(TT\\( dU ' d du > d \ r 1 f dU A\ ( dU >Yi\ 



and in the more restricted case where udx + vdy + wdx is an exact 

 differential, we get 



* (C7l) H^) + VdT)\ + <l>{Ui) {-d^ + ^^-r-dv) =0 - 



