322 Frederick G-uthrie on certain 
be obtained (water at 4° C). Expressed generally, 
§ 3. (a) A cube centimeter of water at 4° C. moves in a 
straight line at right angles to one of its faces at a uniform 
rate of 1 centimeter in 1 second. It traces out a path of unit 
density. This conception is perhaps put into a more useful 
form: (/3) A square centimeter of surf ace having the mass of 1 
gram moves at the rate of 1 centimeter a second ; its path is of 
unit density. The thickness of the surface is here nothing, 
and its density is infinite. More generally: (7) Unit path- 
density is made when a plane surface of area - centimeter, 
1 . n 
weighing ~ gram, moves with uniform velocity. 
Perhaps this is clearer if we imagine an infinite series of 
square centimeter-gram-surfaces following one another at 
equal intervals of 1 centimeter apart, and moving at any velo- 
city along a straight path at right angles to them. Then any 
cubic centimeter of such an infinite path will weigh 1 gram. 
Returning to concrete examples: — 
If Yd denote path-density, 
8 denote density (specific gravity), 
I denote the thickness in direction of motion (centimeter), 
r denote rate (centimeter-second); 
then Pd=^* (1) 
§ 4. The few following obvious deductions may serve as 
illustrations. Td = S when -=1; that is, when the space 
passed over by any point of the moving mass in a unit of 
time is the same as the length of the matter which passes 
through that point when the point is at rest. In other words, 
there must be no gaps. Moving continuous matter has a 
path of the same density as the matter itself. 
§ 5. If the mass be at rest, the expression (1) becomes 
Yd= -7T — , which, though of ambiguous form, here means 
that ~Pd=8. The two O's are equal, having been derived from 
rational diminution of I and r. The same is true if r and I 
are both infinite. 
It is clear, but must be especially noted, That tlie ratio 
between the path-densities Yd ai and Yd a ., is the same whether 
those paths are generated by the single transit {in an eternity} of 
tlie masses Ai and A 2 moving at the rates i\ and r 2 respectively: 
