of determining Stream Lines and Equipotentials. 255 
Then as dn, dco, and dr are in perpendicular directions 
they are independent and 
which reduces to ^— = sr— + a function of r only. 
a- oa) J 
Therefore 
o^ 2 dHo&v \dz 'd&> 'dz"dy)'da) 
B&> 2 d~ 'B&) \d?7 / dot) 2 1 
Again, 
^— =5— .H- + 5- .HI =5-.5- +a function of 
d(f> o<» 09 0*7 09 2w a&> ? , ofi iy # 
. Since 
d?7 length of turn of screw e , . p , 
— = — 9 = = a function ot ■?• only. 
d<j) 2tt 
And so 
^cf> 2 ~ 27r'^<p\d(o )~ 2TrYd(j>'~da)^ 'd<t>''dy)'d(» 
V__ B 2 V 
~ + 47T 2, Bft) 2, 
Now substitute these values of -=r-=- and ^—r , in the ex- 
0~ o^ - 
pression for V 2 Y in cylindrical coordinates, and we have 
which contains only two coordinates r and a>. So that if we 
make V 2 V=/(V, r, g>) over any surface the same will be 
true throughout the whole region filled by the screw-threads 
passing through the surface, provided that ^— is such as to 
make ^— s* + - ^ — constant along every guiding screw. 
qt r or 
One way of satisfying this is to make V increase by the same 
amount per turn of the screw, along everv screw-thread and 
2y ^v 
so that ^—f and ^— are both constant along every guiding- 
screw. 
T2 
