Proceedings of the British Association. 121 



as it has been described in the Philosophical Transactions for 1840, 

 it is unnecessary to do more than to point out the principle on which 

 it acts. The speculum is made to revolve very slowly, while the 

 polishing tool is drawn backwards and forwards by one excentric or 

 crank, and from side to side, slowly by another. The polishing tool is 

 connected with the excentrics by a ring, which fits it loosely, so 

 as to permit it to revolve, deriving its rotatory motion from the 

 speculum, but revolving much more slowly. It is counterpoised, so 

 that it may be made sufficiently stiff, and yet press lightly on the 

 speculum ; the pressure being about one pound for every circular 

 superficial foot. The motions of this machine are relatively so adjusted 

 that the focal length of the speculum during the polishing process, 

 or towards the lateral end of it, shall be gradually becoming 

 slightly longer, and the figure will depend in a great measure upon 

 the rapidity with which this increase in the focal length takes place. 

 It will be evident that a surface, spherical originally, will cease to be 

 so, if, while subjected to the action of the polisher, it is in a con- 

 tinual state of transition from a shorter to a longer focus ; in fact, 

 during no instant of time will it be actually spherical, but some 

 curve, differing a little from the sphere, and which may be made 

 to approach the parabola, provided it be possible in practice to give 

 effect to certain conditions. An immense number of experiments, 

 where the results were carefully registered, eventually established an 

 empirical formula, which affords at present very good practical 

 results, and may hereafter, perhaps, be considerably improved. In 

 fact, when the stroke of the first excentric is one-third the diameter 

 of the speculum, and that of the second excentric is such as to 

 produce a lateral motion of the bar which moves the polisher, 

 measured on the edge of the tank, equal to 27, the diameter of 

 the speculum, or referred to the centre of the polisher, of 1.7, 

 the figure will be nearly parabolic. The velocity and direction 

 of the motions which produce the necessary friction being adjusted 

 in due proportion by the arrangements of the machine, and the tem- 

 perature of the speculum being kept uniform by the water in which 

 it is immersed, there remains still other conditions, which are essen- 

 tial to the production of the required result. The process of polish- 

 ing differs very essentially from that of grinding : in the latter, the 



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