38 president's address — section a. 



irrotational motion. The porosity is not the same every- 

 where. Take as standard substances one in which the 

 amount of fluid that fills its pores is of unit mass, the porosity 

 being independent of du-ection. Then, in a substance whose 

 porosity or " permeability " is fx, the mass of fluid filling unit 

 volume is jx. If the velocity of the fluid be v, then u is a 



/7 O 



differential of a velocity potential Q, and is equal to — -5— : this 



follows from the fact that the motion is irrotational. There 

 will be lines and tubes of flow : along any tube, even if it cross 

 a boundary between two parts of the solid in which jx is 

 different, the amount of flow across any section is constant, 

 i.e., the flux of fiv is constant : for fxv is the amount that 

 crosses unit area per second. Also, since the fluid is incom- 

 pressible, V* ^ = 0. If, further, we suppose that there are 

 fixed in the fluid certain sources of liquid, positive or negative, 

 of whose nature we have information, like that which we had 

 of the sources in the other theories, it is evident that we have 

 the same mathematical problem as before. In some of its 

 details this analogue resembles very closely the theory of 

 magnetism ; and it was on this account that Sir William 

 Thomson examined it. For, if we drop the somewhat un- 

 natural idea of sources and sinks of fluid, and suppose that 

 the whole of space is under consideration, liquid being fi-ee to 

 move through every part of it, more or less, according to the 

 value of fi, then we have the exact analogue of the magnetic 

 field due to electric currents, the velocity at any point being 

 the analogue of the magnetic force. 



It is difficult, however, to represent hysteresis by this 

 analogy. Take a simple case : consider a current running 

 round a single loop of wire. The analogue of this is a 

 motion of the fluid through the loop, round the outside, 

 and back again, the velocity potential of the motion being 

 equal to 4 tt x current. If there be a lump of iron near, the 

 velocity potential will be just the same, but the fines of flow 

 will be altered in number and position. Now, if in the 

 magnetic problem an equal and opposite current be started 

 the field will not wholly cease to exist : there will still remain 

 a field, due to the magnetism retained in the iron. In the 

 other case — the hydrokinetic analogue — if an equal and 

 opposite velocity potential be started into existence, it will 

 destroy the motion altogether. Consequently we cannot, at 

 any rate with a frictionless fluid, represent any of the effects 

 of permanent magnetism. And, again, it does not seem 



