of Edinburgh, Session 1873-74 355 
show the manner in which the resistance varies with the inclina- 
tion of the vanes to the plane of rotation. In these the abscissa 
denote the sines of the angles of incidence (the angle of incidence 
being the angle which the vanes make with the plane of rotation), 
and the ordinates resistances. 
The equation which satisfies these curves, is — 
Sin 3 ^ = R x 0 
where i, the angle of incidence, R, the resistance, and 0, a constant. 
This being shown by a dotted line, found by calculation from 
this formula, and which almost coincides with the curve found 
from the experiments. This clearly proves that the resistance 
varies as the cube of the sine of the angle of incidence. 
The curves C x and 0 2 and the points 0 X , C 2 on the curves of areas 
represent the effects due to concavity of the vanes ; from which we 
conclude, that a certain amount of concavity offers a greater resist- 
ance than the same area, and configuration of a plane surface. 
But, on comparing the greater with the less concave surface, there 
appears to be little or no difference within the limits of these ex- 
periments. This appears to be due to the manner in which the 
particles act upon the surface. First, in comparing the concave 
surface with a plane surface of the same area, we find that the 
concave vane offers most resistance. This may be accounted for, by 
imagining a certain quantity of the particles to be caught in, as it 
were, in front of the vanes, and consequently forming a denser me- 
dium ; this extra dense medium being continually kept up in front, 
while the vanes are in motion. This overcrowding of space has a 
tendency to prevent the particles from moving past the perimeter of 
the vanes with the same ease, and consequently retards the apparatus. 
Again, by comparing the less with the more concave, we would 
at first sight conclude that this was simply an amplified case of 
the foregoing ; but here we have something to balance the extra 
dense medium in front, viz., the action of the particles of the con- 
vex surface behind. Their action may be said to be analogous to 
the action of the water closing in at a ship’s stern ; and, therefore, 
tends to impel the surface forward, and in that way diminish the 
effects of any resistance due to the extra concavity in front. So 
that, looking at the matter in this light, we should conclude that 
2 z 
VOL. VIII. 
