252 Dynamics. 



may be remarked, moreover, that the irregularity from this 

 cause, is rendered for common purposes altogether inconsidera- 

 ble, by making the pendulum very heavy, and the arcs of vi- 

 bration very small. 



374. (2.) A much more important source of error, in the rate 

 of going of common clocks, is to be referred to changes in the 

 actual length of the pendulum arising from heat and cold. A 

 brass pendulum rod, for instance, has its length increased about 

 two hundredths of an inch for a change of temperature of 30 of 

 Fahrenheit's thermometer. This would seem to be a small 

 quantity ; yet as it is continually exerting an influence, the accu- 

 mulated effect in the course of 24 hours or 86400" amounts to 

 more than a third of a minute. The expansion of iron is about 

 f of that of brass. There are some kinds of wood that are sub- 

 ject to very little variation of length, particularly in the direction 

 of the fibres, on account of temperature. Still no substance is 

 entirely free from these changes. The effect of any augmenta- 

 tion or diminution of length in the pendulum may be computed 



346. by means of the principles that have been investigated. 



375. But we can obtain more convenient and sufficiently ex- 

 act formulas for the variation in the rate of the going of a clock, 

 when the changes in the length of the pendulum are very small, 

 as those are which arise from heat and cold, a being the exact 

 length of the seconds pendulum, or that by which the clock 

 would keep correct time, and a' the actual length, as affected by 

 heat and cold, if we put n for the number of oscillations in a day, 

 performed by the former, n' for the corresponding number of the 

 latter, and tf for the time of this latter, we shall have 



346. 



also 



348. 



arcs whether longer or shorter, would be described in the same time. 

 But the practical difficulties attending all the methods hitherto pro- 

 posed, are such as to occasion errors, that more than compensate for 

 the theoretical advantages to be derived from a cycloidal motion. 



