184 BALANCE OF THE CHRONOMETER. 



mate expression m : r z for the moment of inertia of AB, we may 

 employ a more accurate one involving the quantities e and e'; 

 the approximate expression is sufficiently accurate for the deter- 

 mination of A/. 



The adjustment for compensation is effected by shifting the 

 weight m 2 along AB ; that is, by altering the value of <j> until 

 (6) is fulfilled. On the hypothesis we have made, the value 

 of / for t = is not affected by the change of <f>. 



In determining the approximate expressions (5) and (6), we 

 have neglected all terms in which A/z, occurs not divided by T ; 

 all terms involving A/t multiplied by e or e'; all terms into 

 which any power of /t or // enters. In consequence of the 

 last restriction t can only rise to the first power in the result. 

 If this were absolutely correct it would follow that, if the 

 compensation were effected for a particular value of 2, it would 

 subsist accurately for all values of t. For instance, if we 

 give t equal values, positive and negative, the decrease of / 

 in the one case ought to be equal to its increase in the other. 

 But when t is considerable, it is found that there is a sensible 

 deviation from this result; and, assuming that the expression 

 for Ae does not in any perceptible manner involve powers of t, 

 it follows that that of A/ must do so. Any term involving f 

 (and, a fortiori, any higher powers of that quantity), must be 

 very small, since t always occurs multiplied by /A or ///; but it 

 may, nevertheless, sensibly affect the chronometer's daily rate. 

 On the usual construction, the balance oscillates 216,000 times 

 in twenty-four hours. Consequently a very slight change in the 

 moment of inertia of the balance will become perceptible in 

 that period. 



In order to obviate the consequent error, it has been proposed 

 by Mr Dent, a distinguished chronometer-maker of the present 



