266 
ZAHM. 
what too small. Possibly, also, the values of / given in table 
IV for short planes should be slightly increased. 
It should be remarked that the minimum resistance given 
above is such only for the symmetrical shapes in question, 
but not necessarily a minimum for all possible shapes having 
the same major section. In fact, when a five-caliber bow, 
shown by the dotted line in figure 7, was combined with a 
fifty-caliber stern, the resistance was much diminished, and 
it was found incidentally that the ratio of the resistance of a 
good model to that of its major section can be made less 
than one part in eight. What the ratio may be for the 
shape of least possible resistance has not been ascertained. 
Similar experiments were made with spindles having the 
outline shown in figure 8, and with like results. These are 
Fig. 8. —Symmetrical Ogival Spindle of Minimum Resistance. 
still unfinished ; but it may be mentioned, in passing, that 
the frictional effect is very manifest. The total resistance 
of a symmetrical spindle having such outline is again half 
friction, and has its minimum value in a model of about 
twelve calibers, for which the length is nearly seven times 
the major diameter—a relation given by Pankine for well- 
formed ships. A still less resistance is found when a two- 
caliber bow, shown dotted in figure 8, is combined with a 
twelve-caliber stern, in which case the length is about five 
times the major diameter. The ratio of the resistances of 
the spindle and its major section has been reduced to about 
one part in eight. What the smallest possible ratio may be 
for a given velocity has still to be ascertained and may well 
form the object of a special research. 
The foregoing examples suffice to indicate the importance 
of the friction term in the general equations of aerody¬ 
namics. We may now notice its bearing on problems of 
