ATMOSPHERIC FRICTION. 
263 
having the form of cross-section shown in figure 6. The 
head resistance proper may be written equal to that of the 
major section, taken normal to the velocity multiplied by 
the sign of half the angle of the edge of the post. Thus 
R 1 — csina, 
in which c is the resistance of the major section and « is the 
angle abd. Again, the skin-friction resolved parallel to the 
velocity is 
R 2 =2ff,ds . J = 2 ff,dx, 
in which f s is the coefficient of friction for the element of 
surface, and dx is an element of the width a b. Hence the 
total resistance may be written 
R — c sin a -fi 2 fx, 
in which f is the average friction per unit surface. 
A glance at the above equation reveals its chief features. 
For x equal to zero, the second term vanishes, and the first 
becomes 
R= c, 
which is the normal resistance of the major section. For x 
very large the first term is negligible, and there remains 
r = 2 m 
which is the formula for a simple plane moving edgewise. 
Thus the total resistance is comparatively large when x 
equals zero; then becomes smaller and smaller till a mini¬ 
mum is reached, and finally continuously larger as x goes 
on increasing. The width giving a minimum resistance is, 
of course, obtained by placing the derivative of R equal to 
zero and solving for x. 
What has been said of this particular shape is true of all 
the figures of a family in which the major cross-section is 
kept constant while the length varies. There is some length 
for which the resistance is a minimum, and beyond that the 
38—Bull. Phil. Soc., Wash., Vol. 14. 
