BEST PROPORTION FOR DISTANCES OF MAGNETS. 67 



as that of S at distance 1 : and when we find that the 

 Earth's magnet-power is , x that of A, it means that 



A 

 the Earth's action on 8' is j- x the action of S upon S' 



at distance 1. 



30. Investigation of the most advantageous propor- 

 tion of the two distances of the disturbing magnet from 

 the disturbed needle. 



If c 2 were very nearly = c t , the two equations of last 

 Article would be so nearly alike that the divisors giving 

 the results of the equations would be small, and the 

 possible errors in the measures of the angles 6 or <j> 

 would produce enormous errors in the values found for 

 the two unknown quantities. If c 2 were very large, the 

 corresponding deviation would be very small, and the 

 possible errors would bear a large proportion to the 

 deviation. There is therefore a more advantageous pro- 

 portion of c 2 to c v which it is our object now to ascertain. 



The equations being of this form, 



where the actual error of D l may be E^ and the actual 

 error of D 2 may be E 2 ; we find, by elimination of #, 



(r ~ 3 r ~ 5 r ~ s r ~ 3> l r c ~ 5 D c ~ 5 D 

 ( c i -C 2 ' ~ | <? 2 ; x C 2 **% c i L'zy 



and 



subject to the actual error, 



x c ~ 5 E (c ~ 3 r ~ 5 r " 5 r "" 3> > x c ~ 

 xc 2 .&! (c t ,c u c x .c 2 , x c t 



52 



(c ~ 3 c ~ 5 r ~ s r 

 i .c c .c 



