A FEW QUESTIONS IN HYDRODYNAMICS. 
95 
regular polygons having more and more sides? To lend 
plausibility to the surmise that it can be, I write out two 
terms from the expansion of the first solution, three from 
that of the second, and four or more from that of the third, 
thus obtaining 
cos ltd 
cos lid 
[> 
l 1 
£ 2 + y 2 u (X 2 4- y 2 ) 2 ~1 
2 2 2 2 4 2 J’ 
cos lid 1 — l 
D 
«■ + V 2 
n Q 2 + y'J __ 76 o 2 + yy 
L 2 2 4 2 L 2 2 4 2 6 2 
+ 
These show at once that the contour lines become circles in 
the limit as the origin is approached. 
The areas for which exact solutions have been obtained 
are few in number. If a given area does not closely resem¬ 
ble one having a known solution, little can at present be 
inferred with certainty concerning its period or its motion. 
And it may be noted that although an exact solution for a 
given area has been discovered, the complete discussion of 
the possible modes of oscillation involves questions in the 
theory of numbers. This remark will be readily understood 
upon consulting Lame’s Lemons sur la Theorie Mathematique 
de 1’filasticite des corps solides or Riemann’s Partielle Dif- 
ferentialgleichungen. 
When the depth of the body of water in which long-wave 
motion takes place is variable, there is still no particular 
difficulty in putting into mathematical language the re¬ 
quirements of its free oscillation. For, the dynamical equa¬ 
tions, before being combined with the equation of conti¬ 
nuity, are the same as for the case of uniform depth, and 
the equation of the containing surface or bed is supposed to 
be given. Several problems of this class are worked out by 
Lamb in his Hydrodynamics. While it might not be a dif¬ 
ficult matter to state problems concerning the free oscillation 
of a body of water like one of the Great Lakes, the difficulty 
