VOL. LXXXIV.] PHILOSOPHICAL TRANSACTIONS. 339 



of the materials of which the rod cd consists, and in a direct ratio of the ex- 

 pansile power of the materials which connect e and g. That is, supposing that 

 it was taken in the inverse ratio of the inferior expansile power of cd, then it 

 would be at g ; but to counteract the expansile power of the materials ge, it 

 must be somewhat lower at i. 



If then the proportion of the expansion of the materials of the rod ab, or gb, 

 the rod cd and the materials which connected ae or ge were known, and the 

 length of a pendulum swinging any proportion of time, in that case the distance 

 and perpendicular height between ge and ie might be taken at once, and a pen- 

 dulum might be always made which would always be of the same length, and 

 therefore swing equal arches in equal times. But these not being perfectly 

 known, and it being extremely difficult, if not impossible, to measure off length 

 perfectly, it is necessary to have the power of varying the distances and perpen- 

 dicular height between i and e, a and E, or h and e, so that it may be found 

 from trial whether these fixed points i. A, or h, be properly taken. For if, on 

 constructing a pendulum on these principles, either of the fixed points, accord- 

 ing to the circumstances, i, A or H be placed too high or too low, then the 

 pendulum will vary in its length, and of consequence swing different times in 

 different degrees of heat. 



That is to say, suppose the case be taken where the rod bg is made of ma- 

 terials which expand more than cd, if instead of rendering i the fixed point, it 

 is made g, nearer to b, then the point c will not be raised sufficiently above the 

 point e, the pendulum will become longer by heat, and make fewer vibrations in 

 a given time. But if the fixed point in this case be brought lower than i, to 

 the point l, then the point c will be raised higher when the whole is expanded 

 by heat from the point e, ed will be rendered shorter, more vibrations will be 

 performed in a given time, and e contra. Therefore if there be a power in the 

 apparatus of altering the fixed point i, if found too high or too low by experi- 

 ment, we shall be able to find out a true point, and make an adjustment accord- 

 ingly. That is, if heat occasions a pendulum to make fewer vibrations, the 

 fixed point is too high ; if on the contrary it makes too many vibrations, then 

 the fixed point will be too low. 



I come now to show how these principles may be applied in structure. 



As it is more convenient in practice to have the fixed point higher than a, 

 which is equal in height to d the bottom of the pendulum, a substance should 

 be chosen for the rod ab which expands and contracts more by heat and cold 

 than the matter of which the rod cd consists, so that the fixed point should be 

 at G, if the materials connecting a and e were incapable of being expanded or 

 contracted by heat or cold: but if their expansion and contraction is to be com- 

 pensated for, the fixed point will be at i, as has already been shown. 



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