662 Proceedings of the Royal Society of Edinburgh. [Sess. 
been so designed as to be equally serviceable with a variety of different 
wheels. There is therefore no reason why a number of such wheels 
should not be provided, and employed as occasion may require. 
Fig. 4 suggests another type of wheel, shown in fig. 7, consisting of two 
'plane circular discs fixed on a shaft at a small distance apart, parallel to 
each other, and having the same number of equal sectors cut out of each. 
As in the analogous case of the paddle-wheel of fig. 4, 
E = <r + | e | ,* 
Fig. 7. 
but here e is not given by the equation which will be deduced below for the 
e of the previous cases ; but, as may readily be shown, by the equation, 
, 2?/c tan a 
tan e = - — — - , 
y z + x z sec z a - c z tarn a 
where x and y are the coordinates of the ray of light considered, referred 
respectively to a horizontal axis through 0 (which here is not the point of 
attachment of the vanes to the axle, but the point of the axle midway 
between the discs) perpendicular to the ray, and to a vertical axis through 
0 ; and where c is half the distance between the discs. Now, it will be 
found in this formula — and herein it differs from the other referred to — 
that a small change in the value of either x or y will cause a considerable 
change in the value of e\ that is to say, in this case the values of the 
eclipse-angles for the different rays of the beam of light will differ too much 
among themselves for such a type of paddle-wheel to be useful in practice. 
It may be well at this 'point to emphasise the fact that the mathematical 
investigation which follows has been worked out chiefly for the purpose of 
* <r' being the eclipse-angle of one of the actual vanes, and e being the eclipse-angle of one 
of the imaginary vanes, each of which must in this case be considered as consisting of a 
rectangle connecting the edge of a front vane to the corresponding edge of the vane behind it. 
