The beautiful photographs obtained by Professor Hele-Shaw of the stream lines in a liquid flowing between two close parallel walls are of very great interest, because they afford a complete graphical solution, experimentally obtained, of a problem which, from its complexity, baffles the mathematician, except in a few simple cases.

In the experimental arrangement liquid is forced between close parallel plane walls past an obstacle of any form, and the conditions chosen are such that whether from closeness of the walls, or slowness of the motion, or high viscosity of the liquid, or from a combination of these circumstances, the flow is regular, and the effects of inertia disappear, the viscosity dominating everything. I propose to show that under these conditions the stream lines are identical with the theoretical stream lines belonging to the steady motion of a perfect (i.e., absolutely inviscid) liquid flowing past an infinitely long rod, a section of which is represented by the obstacle between the parallel walls which confine the viscous liquid.

Take first the case of the steady flow of a viscous liquid between close parallel walls. Refer the fluid to rectangular axes, the origin being taken midway between the confining planes, and the axis of z being perpendicular to the walls. As the effects of inertia are altogether dominated by the viscosity, the terms in the equations of motion which involve products of the components of the velocity and their differential coefficients may be neglected.