BALANCE OF THE CHRONOMETER. 181 



sents a common form of the chronometer balance. The arc AB, 

 which carries a weight at 0, is formed of two concentric laminae 

 of different kinds of metal, the outer lamina being the most ex- 

 pansible. These two laminae are securely united in their whole 

 length, so that an increase of temperature necessarily distorts 

 the arc AB into some form like AB'. Similarly for ab. Con- 

 trary effects are produced by a decrease of temperature. Thus, 

 the moment of inertia I decreases as t increases ; as e also does. 



/> 



And thus we are enabled, by suitable adjustments, to make j , 



at least approximately, constant. 



It would, I believe, be impossible, without some hypothesis, 

 to determine the form which AB assumes under the influence 

 of a change of temperature. The following suppositions are 

 probably sufficiently near the truth to be applicable when the 

 variations of t are not excessive. 



Let us suppose the laminse to be cylindrical and concentric, 

 and bounded by four plane surfaces, two of which are perpen- 

 dicular to the axis of the cylinder, while the other two, which 

 form the boundaries at A and B 7 pass through the axis. These 

 conditions being fulfilled, whatever the value of t may be, it is 

 clear that the variation of form can depend on two elements 

 only, namely, the radius of the cylinder, and the angle which 

 AB subtends at its centre. To determine these, we assume 

 that the middle filament of each lamina expands as it would do 

 if free. 



In the normal state, let 2e, 2e' be the thicknesses of the outer 

 and inner laminae respectively, r the radius of the boundary of 

 the two laminee, //., fju the coefficients of expansibility of the 

 outer and inner laminae (yit > //), 6 the angle subtended at the 

 centre. 



The radii of the middle filaments are, therefore, r + e, r e' ; 

 let their lengths be I and Z', then 



For an increase of temperature t, let r and 6 become TI and % : 

 then we shall have 



= r - 



and e' being so small that their variations may be neglected. 



