XX INTRODUCTION. 



The values of m X and being known, if m X = A and _ = B, 



X -A- 



--^-B 



To determine the value of K, the magnet, stirrup, &c., is vibrated a second time, with the 

 addition of a metal ring, whose internal and external diameters and weight are correctly known. 

 Denoting the two radii (in decimals of a foot) by r and r n and the weight (in troy grains) by 

 W, the moment of inertia of the ring K' is 



K' = i (r 2 + r , 2 ) W ; 



then if T represent the time of vibration of the magnet without the ring, and T' the time of 

 vibration with the addition of the ring, the moment of inertia of the magnet and stirrup in the 

 vibrations without the ring is expressed by the formula 



/ T 2 \ 



V TT/ I _ _ 1 



V/jya T 2 / 



It is necessary to correct the value of T for the force of torsion of the suspension thread, 

 the rate of the chronometer used in the observations, the arc of vibration, and for the changes 

 in the dimensions or form of the suspended mass produced by changes of temperature. The 



rate of the chronometer denoted by x \ _ l S ai . nm g ( j g supposed to be known ; the semi- 



arcs of the vibration at commencement and end of the series in parts of radius d and d' are noted 

 from the scale, and the ratio of the torsion force of the suspension fibre to the earth's directive 

 force is obtained by turning the torsion circle through two or more large angles, and noting the 

 corresponding differences of scale reading. If the coefficient of the torsion force H, the 

 earth's directive force F, the mean of the angles = w, and that of the differences of scale 

 reading reduced to angular value == u, then 



H - 



F 10 



a a' 



The correction for the arc of vibration is found by the formula = a 2 dd' xO.000072722 2 , 



16 



in which a denotes the angular value of a scale division, d and d' the semi-arcs of vibration 

 in scale divisions, and the last correction, viz : for change in the dimensions of the suspended 

 bar, consists in multiplying the value of K by the quantity 



1 + 2e(f 0, 



t' denoting the actual temperature of the magnet, t the temperature corresponding to the time 

 of the original observations, and e, the coefficient of the dilatation of steel for 1 degree Fah- 

 renheit. The numerical value of & is 0.000068. 



In out-door observations, as ours always were, the temperature is rarely the same at the 

 vibration and deflection experiments. During the former, the magnet is enclosed in a nearly 

 air-tight box of wood, within which the temperature is quite steady, and at those of deflection 

 the same bar is exposed to the fluctuations of every passing current as well as to horary changes. 

 As the magnetic moment of the deflector varies with every rise or fall of the thermometer, if 

 the temperature at the time of the experiments of vibration differs from that during the experi- 

 ments of deflection, a correction is necessary on this account. The mode in which the tempera- 

 ture coefficient, represented by q, was determined, will be detailed farther on ; the value for each 



