METHOD OF GAUSS. 605 



Let us assume that, the distances R and R' being exactly known, 

 the errors made in measuring the angles a and a! are da and da!-, the 

 corresponding error on / will be 



da! da 

 ~ 



Other things being equal, this error is the least possible when the 

 product (i -/> 2 )/o 3 is a maximum that is, when 



/o 2 = ~, or = 1-29 about. 

 5 ^ 



With this value of p, the angles a and a are almost in the ratio 

 of 2 to i, and we have sensibly p 3 a = a', which gives 



In order that the error dp be less than o'ooi, the relative errors 



da! da 



and , which we shall suppose equal and opposite, are less than 

 a a 



0*0002. If the angle a is 10, or 600', the probable error da must 

 then be less than 4"; these are conditions which are difficult to 

 realise in practice. 



The relative error on the determination of the ratio is equal 



xi 



to dp, and the corresponding relative error on the value of the 

 component is half less. From this it will be seen that the principal 

 difficulty consists in this term of correction. 



In permanent observatories it is better to use a special apparatus 

 for this kind of experiment. 



1177. For travelling observations, magnetic theodolites have 

 special arrangements, by which the field of the magnet may be 

 compared with the terrestrial field. In Brunner's compass (Fig. 240) 

 a lateral rod on the box supports two stirrups R and R', on which 

 a bar can be placed at two different distances, to make it act on 

 another magnet, of half the length, and provided with copper ends. 



The microscope being sighted for the movable magnet, the de- 

 flecting bar is placed on a support, and the instrument is turned so 

 that the sighting of the microscope is unchanged ; the same obser- 

 vation is then made after having reversed the deflecting bar. The 



