196 Mr J. Brill, On Conjugate Functions [Feb. 27, 
is expressible as a function of a complex variable. Hence we see 
ieee 
is also a solution of Laplace’s equation. 
It seems highly probable that this process may be repeated 
again and again, and thus we should be furnished with a second 
series of expressions of a general type capable of being expressed 
as functions of a complex variable. 
The modified form of Lamé’s equation given in this article 
was used in reducing the last expression of Art. 5 
5. Suppose that we are considering a case of irrotational 
motion of an incompressible fluid. Let p be the pressure and h 
the velocity at any point of the fluid, € the velocity potential, 
n the current function, V the potential of the external forces, 
and 6 the density of the fluid. We have the equation 
Pp 05 oo 
5 ever AE +5h =) 
from which we easily deduce 
EG +m) 15 ato ae (*) = (=) = Le 
: ae) Se Dee || == |) Se Hie +(-) =0, 
oe ot 0& \p, On \pe Px Pe 
where p, and p, are the respective radii of curvature of the curves 
£=const. and »=const. at their point of intersection. Making 
use of the THoETPEd form of Lamé’s equation, we deduce 
fe o° = {5 4V4 tt + 2i(-) 4 (-)| 7 
If the fluid be homogeneous, and V satisfy Laplace’s equation, this 
becomes 
O"p sat | EN Ne 
+24} 2+) -0 
5 Toe an)” \\p/ * \p, 
From this it follows that 
(ape) 8 (ape ap) =O 
from which we easily deduce 
