182 BALANCE OF THE CHRONOMETER. 



r t + e I \+p* 



Hence vl?"ri*7ft' 



and therefore 



which, as //, and // are very small, is approximately 



When = 0, r- 6 '=(6 + e') r 



Consequently, for a first approximation, 



(1), 



where Ar = ^ r, A/A = ft //, and T = e + 

 Again T^ = I - 1 1 + (pi - /*T) . 



When * = 0, rd = 1-1'-, 



and therefore rA# = (/*? - /x7') ^. 



But fd - pi' = r0&fjL + (fie - tie] 0, 



and the last term is negligible. Therefore 





(2) 



We distinctly perceive from (1) and (2) why the effects of 

 distortion are so considerable in comparison with those of simple 

 expansion ; it is because the expressions of Ar and A0 have the 

 small quantity T in the denominator. AB becomes a larger arc 

 of a smaller circle. 



To apply these results : we suppose that when t = the 

 centre of AB coincides with the central point 0; and AB, 

 being securely fastened at A, continues perpendicular at that 

 point to the line OA, consequently its centre remains in that 

 line. Let 0' be its new position, then 00'= Ar. Ifwjbe 

 the mass of AB, its moment of inertia about was ??2r 8 ; 



