90 Mr. Cecit on a Machine for Grinding 
The mirror must be carried backwards and forwards across 
the centre of the leaden tool with an equal length of stroke. The 
force which produces the rectilineal motion should be applied 
horizontally, in a plane nearly coinciding with the surface of the 
mirror, so that the friction on the surface and the force which 
overcomes it may destroy each other, both being nearly in the 
same plane. 
During the whole process, both the mirror and the leaden tool 
should be made to revolve uniformly, but very slowly, about their 
own axes, in opposite directions; that all parts may be ground 
equally. It is also desirable that the angular velocity of the mirror 
and of the leaden tool should not be the same, but one double of 
the other, and in opposite directions. For if the mirror be some- 
what elliptical, having its opposite diameters of different curvature, 
the leaden tool being also similarly affected, by this contrivance 
the diameters which are least adapted to each other will be ground 
together, and will approximate rapidly to an equal curvature: 
this would never be the case if the angular velocities were equal*. 
Opposite motions are preferable as producing the greatest relative 
motion with the least absolute; that is, with the least friction 
and inertia. It is also very convenient that the mirror should be 
capable of being easily taken up, to be cleaned or examined ; 
* Tig. 1. Let AB, CD, be the major and minor axes of the leaden tool. 
— ab, cd mirror. 
MUN the direction of the stroke. 
The upper figure may represent the position which results from grinding ; where the major axis 
of the mirror coincides with the major axis of the leaden tool, and both with the direction of 
the stroke. 
First, let the angular velocities be equal and opposite; and let AB, ah, revolve 90°. 
Then the second figure represents the new situation, in which the axes minores coincide, and rub 
length-ways upon each other; without any tendency to correct the elliptical form. But if the 
inner wheel revolve twice as fast as the other, the lowest figure represents the new situation ; 
in which the axis major of the small wheel, (with its extremities inverted), rubs length-ways 
upon the axis minor of the large wheel; whereby they will speedily be reduced to a mean and 
common curvature. 
