64 Mr Brill, On Families of Curves. [Feb. 24, 



Suppose that we have two functions u and v, which are con- 

 nected by the relations 



du dv , du dv 



~ 5- =cu and 5- + 5- = — cv. 



ox oy oy ox 



From these we easily deduce 



{fx + i dl)^ + iv) = c{u - iv) ' 



dy 



,d_ 

 \dx ". dy 



% ^-)(u — iv) = c(u + iv). 



Thus we have 



g^2 + ^1 (w + iv) = c 2 (u + iv), 



and therefore u and v are both solutions of the equation given 

 above. 



Further, the relations connecting u and v may be written in 

 the form 



These relations show that we may express u and v in terms of 

 two new functions <j> and i/r as follows : 



u = <rf, v = e^, u = -e-^, v = e- d f. 

 oy ox ox oy 



From this it follows that 



2cx d_±_d± d ^±__d± 

 ox oy oy ox 



Another good example is given by cases of irrotational fluid 

 motion symmetrical with respect to an axis. In this case we 

 have a velocity potential (f> and a stream function ty connected. by 

 the relations 



dr dz ' dz dr ' 



These two examples will suffice to show that among the classes 

 of systems discussed in this paper there are some which have 

 applications to problems of interest. 



