3Z- 



= ir.{T — ^r COS. <p), 



or taking T as the standard temperature, and putting g — the length 

 of that arm which is horizontal, the change of each radius from ^ 

 is = — i fsr. cos. ^, and the equation of the deformed circle becomes 



a =:{.(! — ^«T. COS. <p ) 



But this equation holds only for the extremities of the arms ; which 

 however from their superior strength command the limb : they 

 also suffer a little flexure, as the arcs which connect them acquire 

 by expansion a greater length than belongs to the portion of the 

 above curve intercepted between the arms. Rectifying it by the 

 usual method, 



^'^ 16 (l+iiVf '*^* 



^'"•M'sVttf? +''" 



Again the temperature of a differential portion of the arc is 

 (T — r. COS. <p)\ while the arc increases by d(f>, the sum of the 

 temperatures of all its parts increases by this quantity, and therefore 

 the mean temperature of the arc is 



f. (T — T COS. (f).dp T. sin. ip 



f.d(p p ' 



which needs no constant. Therefore the mean expansion of the 

 arc a. intercepted between two arms is — §e. t.(2 cos. {<p — ^a) sin. 

 ^a, while the intercept of s between them is less than its original 

 bulk by 



little more than half the other. Therefore each arc must bend a 

 little, and the difference of the forces required to produce this 



