338 PHILOSOPHICAL TRANSACTIONS. [aNNO 1794. 



than a simple pendulum made of the same materials of which the rod cd consists, 

 and it would be nearly of the same length as if the pendulum had been made of 

 the materials which connect a and e. 



In order then to render ed always of an equal length, some other principle 

 must be employed. Now let ba and cd consist of the same materials ; and the 

 matter conrtecting a and e consist of a substance that expands by heat, and con- 

 tracts by cold, less than the materials of which ab and cd are formed : if the rod 

 AB be brought down to h, and the fixed point be at h, and the points he be con- 

 nected together by the same materials which formerly connected the points ae 

 in the rod eh, take a point k, equal in height with the points a, d, and in the 

 line AD which is parallel to the horizon, as it has already been taken in the con- 

 struction : then the rod ah shall expand, on being heated, in perpendicular height 

 more than the rod hk, and therefore the expansion of ah shall carry the point 

 B, and in consequence the point c, higher than the expansion of the rod hk shall 

 carry the point e ; but if the expansion of the rod ah be as great as the expan- 

 sion of the whole rod he, then the point c will be carried as much higher than 

 the point e, as the point e is carried higher than it was before the heat was ap- 

 plied, and therefore the point e shall beat as great a distance from the point c, as 

 it would have been if the materials connecting ae had been incapable of being 

 altered by heat : and therefore if ed be a pendulum, it will be rendered of the 

 same length. If then the rods ab and cd be of the same materials, and the sub- 

 stance connecting a and e be capable of expanding and contracting less by heat 

 than the matter of the rods ab and cd, then, by adding to the rod ab, a part ah, 

 under the circumstances already described, and making the fixed point at h, we 

 can obtain a pendulum always of equal length : or if the materials of which the 

 rod AB consists be capable of being expanded by heat as much as the materials of 

 which CD consists, together with the expansion of the materials that connect a 

 and E, in that case likewise the point c shall be carried as much higher than the 

 point E, as it would be when they are expanded by heat, as if the materials con- 

 necting A and e had not been affected by heat at all. Or lastly, if we take a rod 

 CB, of materials which expand much more than the expansion of the matter of 

 the rod cd, and the connecting matter of the rod eg by heat, then likewise on 

 the rod gb expanding, it will carry the point b, and consequently the point c, as 

 much higher than the point e, as it would have been if the materials connecting 

 G and E had been incapable of being expanded by heat, and therefore ed will 

 always continue of the same length. The same reasoning will hold in cases of 

 contraction from cold. 



Therefore, if materials be employed in the rod gb, which contract consider- 

 ably more than those which compose the rod cd, then the fixed point o is to be 

 taken at a distance from b, in an inverse ratio to the inferior expansile power 



