But the resistance _R_ indicated on pages 23 and 24 is not the only resistance to propulsion. In addition, we have the frictional resistance of the air sliding along the sail surface. The amount of this resistance is independent of the angle of inclination: it depends directly upon the area of the planes, and in an indirect way on their dimensions in the direction of movement. It also varies nearly with the square of the velocity. At any velocity, then, the addition of this frictional resistance, which does not depend on the angle of inclination, modifies our views as to the desirable angle: and the total resistance reaches a minimum (in proportion to the weight supported) when the angle is about three degrees and the velocity about fifty miles per hour. This is not quite the best condition, however. The skin friction does not vary exactly with the square of the velocity: and when the true law of variation is taken into account, it is found that the _horse-power_ is a minimum at an angle of about five degrees and a speed of about forty miles per hour. The weight supported per horse-power may then be theoretically nearly a hundred pounds: and the frictional resistance is about one-third the direct pressure resistance. This must be regarded as the approximate condition of best effectiveness: not the exact condition, because in arriving at this result we have regarded the sails as square flat planes whereas in reality they are arched and of rectangular form. At the most effective condition, the resistance to propulsion is only about one-tenth the weight supported. Evidently the air is helping the motor. Resistance of Dirigibles If the bow of a balloon were cut off square, its head end resistance would be that given by the rule already cited (page 19): one three-hundredth pound per square foot, multiplied by the square of the velocity. But by pointing the bow an enormous reduction of this pressure is possible. If the head end is a hemisphere (as in the English military dirigible), the reduction is about one-third. If it is a sharp cone, the reduction may be as much as four-fifths. Unless the stern is also tapered, however, there will be a considerable eddy resistance at that point. [Illustration: HEAD END SHAPES] If head end resistance were the only consideration, then for a balloon of given diameter and end shape it would be independent of the length and capacity. The longer the balloon, the better. Again, since the volume of any solid body increases more rapidly than its surface (as the linear dimensions are increased), large balloons would have a distinct advantage over small ones. The smallest dirigible ever built was that of Santos-Dumont, of about 5000 cubic feet.