of determining Stream Lines and Equipotentials. 247 
Now 
TT 
and therefore vanishes ; so that the chequer ratio ~ is a 
function of a and /3 only. 
As has already been stated, to make V 2 « = on the surface 
Y we must have 
which may conveniently be arranged by making 
^- l TT 
where k is a constant, so that the chequer ratio is given as a 
function of position on <y . This is more than sufficient in 
that it makes 
9_/H 1 fl L A 
3£V Ha / 
vanish as well, but the loss of generality involved is found 
not to matter, while the simplicity gained is a great con- 
TT 
venience. Next, because' ^ is independent of 7 it follows 
that on any other surface y x we still have 
H * — I TT 
Hj8 - ^o-tlyo- 
But if \J 2 a=0 is to be satisfied on this second surface we 
must there have 
Ho 
H 
where h x is a second constant. Therefore regarding y as 
fixed and 71 = 7 as movable we have H y = Hy x a function of 
7 onl J- 
But H ro is a function of a and /3 only. 
This relation is equivalently expressed by the two equations 
