660 Proceedings of the Royal Society of Edinburgh. [Sess. 
angle, the latter being positive when the motor is turned counter-clockwise 
about its pivot, looked at from above, and negative when turned clockwise. 
With this convention it is evident that (1) will apply also to the case 
shown in fig. 3, when a and e are negative, provided the equation be 
written 
E= | o- + e | . . . . . . (A') 
Again, we write for a vane of the paddle-wheel shown in fig. 1, 
M (A) 
an equation which affords no new information in itself, but is required 
in connection with the general theory. 
Fig. 4. 
Another type of paddle-wheel closely analogous to the preceding is 
shown in fig. 4. If the reader will imagine the following changes to take 
place in this new type of wheel, he will see how it may be transformed back 
into the type of wheel just considered. Suppose that each edge of each 
vane is connected to the edge of the vane directly behind it by a flat 
triangular piece, e.g. OA connected to OB ; and then suppose that all the 
vanes are subsequently removed, leaving only these triangular pieces. 
They will form a paddle-wheel only differing from that shown in fig. 1 by 
the possession of a greater number of vanes. Fig. 5 shows diagram- 
matically a pair of vanes such as are pictured in fig. 4, when looked at 
from above. In this case the eclipse-angle E caused by the action of 
these two vanes is clearly equal to the eclipse-angle of an imaginary vane 
BOD of the fig. 1 kind, plus the eclipse-angle of the actual vane AB. This 
latter angle of course is constant and essentially positive ; calling it a, and 
calling e the eclipse-angle of the (imaginary) vane BOD, we have 
E = <x' + e . . . . . . (2)' 
When the axle is turned to the left, as in fig. 6, it is still the sum of the 
