676 Proceedings of the Royal Society of Edinburgh. [Sess. 
arrange matters that as t 0 decreases, so also (numerically) does e, and with it 
the variation of t 0 ; thus bringing about the ideal result that, when t 0 is at 
its smallest, the absolute amount of the change to which it is liable is 
simultaneously a minimum. 
For the purpose of finding quantitatively in such a case the necessary 
restriction on the width of the light beam the former discussion is still 
serviceable. 
For positive values of e equation (17), the fundamental equation in the 
present case, is identical with equation (11), the fundamental equation in 
the last case ; and on the other hand, when e is negative, so also are a, p, 
and e, and the argument will be found to proceed mutatis mutandis 
precisely as before, and to lead up to the use of Charts III. and IV. as 
explained above. 
Another restriction on the permissible size of the cross section of the 
beam is due to the evident necessity for having all the rays near enough to 
the paddle-wheel to be duly intercepted by the vanes as they spin round ; 
and this may be called the “ contact condition.” 
§ 5. The " Contact Condition.” 
The following method of discussing the condition applies to all the types 
of paddle-wheel that have been described above. Imagine two plane discs 
of unit radius fastened to the axle of the paddle-wheel perpendicular to the 
axle; the front one having its periphery in contact with the front extremities 
of the vanes ( e.g . with ff in fig. 1) ; and the rear one having its periphery in 
contact with the rear extremities of the vanes {e.g. with g g in fig. 1). Then 
clearly all rays of the beam, that are properly intercepted by the vanes as 
they rotate, will pass through both these imaginary discs, not outside either ; 
or, to put the matter in another way, through the common part of the 
orthogonal projection of the two discs on an imaginary plane through 0 
perpendicular to the beam. That is, the entire beam must be contained 
