61 



let the needle be displaced by a small amount from the posi- 

 tion of equilibrium, — and let the force brought into play by 

 the displacement be just balanced by friction ; then 



f= mR cos Uq^Uq , 



/ denoting the moment of friction. Now, this being constant 

 for a given instrument, cos MqAmo is so likewise : and we have 



COSMqAMo = £> 

 E denoting the value of Amq corresponding to Mo = 0, or the 

 limit of the error due to friction in the natural position of the 

 needle, under the influence of the earth's magnetic force 

 alone. 



To find the error in the value of R, corresponding to A?<u, 

 we have only to differentiate the equation of equilibrium with 

 respect to R and Mqj and we have 



A /2 sin Mo + i? cos Mq AMq = ; 



and, substituting for cosMq Amqj its value above given, 



A-R _ -£ . 

 R sin Mo * 



We see, then, that the relative error in the value of the 

 force resulting from friction, in either part of the process, is 

 inversely as the sine of the angle of deflection ; and that it is, 

 therefore, requisite for accuracy that these angles should be 

 considerable. The angle of deflection may obviously be as 

 large as we please in the first part of the process, where the 

 deflection is caused by a weight; but, in the second, a large 

 deflection can only be produced by a massive magnet, and 

 such a magnet cannot be employed in the first part without 

 impairing the accuracy of the result by the increased friction. 

 The conditions of accuracy required in the two parts of the 

 process are, therefore, incompatible. 



We evade this difficulty by employing the inclinometer 

 for one only (namely, the second) of the two observations, 



