64 Dr. A. F. Zahm on Atmospheric 
The equation R=av" was found to express very accurately 
the resistance of all shapes tested, at speeds from 5 to 
A() feet a second. For normal planes, spheres, cylinders, and 
blunt bodies generally, except very small ones, 2 equals 2, 
very approximately; for thin, tapering bodies n may have 
any value from 2 to 1°85. But in every case, if the form 
and aspect of the model remain fixed, a and n are found to 
remain invariable for all the speeds employed. This was 
manifested by plotting the speed and resistance on logarithmic 
cross-section paper, and observing that the diagram was 
invariably a straight line for all the models tested. 
These conclusions, though apparently valid for the speeds 
employed, which ranged from 5 to 40 feet a second, 
may not all be extended to considerably higher velocities. 
The well-known formula for blunt bodies, R= av’, fails when 
employed for a wide range of speeds, as shown by Duchemin* 
and by the present writer +; the formula R=av? + bv® being 
much more accurate. So, likewise, the equation for friction 
may have its limitations. Even at the ordinary speeds of trans- 
portation, at which the Newtonian formula obtains for blunt 
bodies, not too small, it fails for sharp ones, owing to the 
element of friction. It seems, therefore, that for such speeds 
the resistance might better be expressed as the sum of two 
terms, one giving “the head resistance the other the component 
of surface-friction. Thus 
R=av?+be'™. 
This formula has, indeed, been ed satisfactorily to a 
number of easy shapes, the computations agreeing very 
closely with the data of observation. 
Such were the results obtained in a wind of uniform 
velocity and direction. When, however, the current is tur- 
bulent, @ and n are found to vary considerably. But, since 
the flow of a turbulent wind cannot be specified, the measure- 
ments made in one such current cannot well be applied to 
determine the resistance of a different one. For that reason 
it seemed better to make the measurements in a uniform 
wind, where, moreover, the instruments give steadier 
readings. 
On comparing the above results with those obtained by 
Froude for water, it is found that the equations are very 
similar. The exponents are nearly the same, and the co- 
efficienis are to each other roughly as the densities of air and 
water. Using varnished friction-boards, Froude finds n= 1°85. 
* ‘Tes Lois de la Resistance de |’Air sur les Projectiles.” 
+ “On the Resistance of the Air at Speeds below One Thousand Feet 
a Second,” Phil. Mag. May 1901. 
