OF WHICH IS MOVING ROTATION ALLY AND PART IRROTATIONALLY. 365 
British Association on “ Recent Progress in Hydrodynamics,” Part 1,* “ That the 
results refer to the cyclic motion of the fluid as determined by the supposed distribution 
of magnetic matter, and do not give the most general motion possible.” It appears 
also from Examples I. and III. of this paper that it is not possible to assume arbitrarily 
the distribution of vortex lines, even when it can be shown that the equations of 
motion are satisfied at all points where the fluid is moving rotationally, and then to 
proceed to calculate the irrotational motion by means of the supposed distribution of 
magnetic matter. For in these examples, values of the components of the velocity of 
a rotational motion, satisfying the equations of motion throughout a finite portion of 
the plane of x, y, are found. Thus the distribution of vortex lines, and, therefore, that 
of the supposed magnetic matter over a finite portion of the plane of x, y is known. 
The surfaces that always contain the same vortex filaments are found. Inside one of 
these the supposed magnetic matter is distributed, the current function at external 
points is calculated by Helmholtz’s method, and it is shown that the velocity thence 
deduced is not continuous with the velocity of the rotational motion at the surface, 
which separates the rotationally moving liquid from that moving irrotationally. 
Another way (suggested by Clebsch’s forms) of obtaining the velocity potential will 
be as follows :— 
• 1 / (Py dry 
Calculate the quantity p= ^ 
Treating p as the density of a material distribution inside \.= 0, taking no account 
of the value of p outside the surface \= 0, obtain the potential of this distribution. 
Let the potential inside \ = 0 be y', and outside let it be <£. 
X will, in general, differ from y; first, because y may contain many-valued terms, 
which may be denoted by P, satisfying Laplace’s equation; and, secondly, because 
y—P may be the potential of a distribution of matter, part of which is outside \=0. 
Accordingly, it is necessary to examine whether it is possible to find many-valued 
terms P satisfying Laplace’s equation such that y'-fP = y. 
Then <£+P will be the velocity potential of the irrotational motion, provided that it 
give zero velocity at infinity. 
Example II. of this paper is solved in this manner. It might also have been solved 
by Helmholtz’s method. 
The few illustrations which follow are a first attempt to apply the theory to 
particular cases. 
Example I. treats of the motion of an elliptic vortex cylinder of invariable form 
parallel to one of its axes with arbitrary velocity. The irrotational motion outside 
the cylinder cannot be supposed to extend to an infinite distance. 
Example II. treats of Ktrchhoff’s elliptic vortex cylinder, in which the angular 
velocity of the rotation of the cylinder is a function of the vortex strength, and the 
axes of the elliptic section of the cylinder. 
d? x 
* Report for 1881, Part I., p. 64. 
