1888. ] and Hquipotential Curves. 195 
Thus the second of our two equations becomes 
ae 
ees eh pat 
and therefore 
a (=) 0. (=) 
OE \p,/ On \p,/” 
as is obvious from the expressions for p, and p,. 
From the modified form of Lamé’s equation given above we 
deduce 
Nese aN Ch oh 
we(Q) + Gy nmet ne 
sal ia NBs ai © ae ° On 
therefore 
: Oh oh oy (le, JL) 
2 (log h’— 1S’) = log h + lo ape t an “} 7 s+ 2 =]: 
(log )= leg 8 138 a an as 
which shews that 
Ge Gall By (0 Ye 
log fae opi —*| SBE 
is expressible as a function of a complex variable. Thus we have 
CG Oh oh 
(Gea) 18 jae tae} —° 
from which we easily deduce 
Ue + = ] 
j& oy") ‘ 
If we expand the equation 
or +a). ENS payeka\s 
log (=) + ~) = 0, 
Get ° (py 6 
take logarithms, and make use of all of the expressions in Art. 3, 
we deduce that 
los (52+ 5) ap) +) H 
r 2. 0 ez bs 
— 21 | 3+ tan™ 8 \p,) op, 
ee gee) | 
Oh Oh 
: (oe Yay }= ‘ 
