324 Frederick Guthrie on certain 
right angles to the path. The sectional area of the path is of 
course proportionally less, being sin 6. 
§ 9. Path-mass. — For unit of path-mass it will be conve- 
nient to take the same conditions of motion as before (§ 3), 
namely, a square gram-centimeter moving at the rate of 
1 centimeter in 1 second at right angles to itself. 
If two such plates follow one another in a second, the path- 
densny and the path-mass are both doubled. If they are 
placed edge to edge in one plane and move with the unit 
velocity, their path-mass will be doubled. For the path has 
the unit density but double the volume of that generated by 
the single square. Just as in ordinary mass we have 
M = SV 
(where M = mass, 8=. density, and V = volume), so here we 
have 
P m =VP d . 
Of course, if the path be maintained at uniform density by 
the passage of successive surfaces, the total mass will be infinite. 
But the mass per unit length is proportional to the sectional 
area of the path at right angles to it. And this is the case 
when a finite path is maintained at constant density by the 
to-and-fro motion of the surface or by its orbital return, as 
in § 5. 
§ 10. However a mass of matter may rotate, however irre- 
gular it may be in shape or density, however it may change 
its shape or density, and whatever may be its velocity or change 
of velocity, its total path-mass is constant for the same time- 
interval provided only its mass remains the same. 
§ 11. If the earth's orbit be 300,000,000 kilometers in dia- 
meter (a little more than 186,000,000 miles) and its diameter 
be 8000 miles, the height of a cylinder having the same volume 
as the earth and the same diameter is 5333 miles, or 8580 - 797 
kilometers. It therefore appears that the mean orbit-density 
of the earth is 7 - 761 times the specific gravity of the earth, 
or, say, about 39 or 40. 
§ 12. If a single moving atom were enclosed in a box of 
fixed internal dimensions, the mean density of the gas con- 
structed by it would be constant, however the atom might 
move. But if the atom move to and fro between opposite 
walls with acceleration, say harmonically, its path will be 
denser at its extremities and densest at the walls according to 
the law of sines, or as though the material area between a 
semicircle and its tangent had been compressed upon the tan- 
gent (fig. 3). 
§ 13. The increased path-density caused by the retardation 
