Determination of the Earth's Horizontal Magnetic Force. 391 



mental formula? being granted — how largely q and q f are in error : but 

 this is an exceptional occurrence, especially in field observations. To 

 consider all possible mean temperatures seemed unnecessary, and I 

 thus confined my attention to the three cases 



(t + t')/2 = 0, = 15° C., = 30° C.; 



corresponding to which, 



8q + (t + t')8q' = 8q, = 8(q + 30q'), =? S(^ + 60^'), 



In other words, I found the difference between the two observed values 

 of the three quantities q, q + SOq' and q + 60(/, treated independently, 

 and the corresponding three independent probable errors. 



The results, derived from seventy magnets, were as follows : — 



Quantity. 10V 10 6 (<? + 3Qq'). 10 6 (q + GOq'). 



Mean semi-difference 11'5 7*8 14*1 



Mean probable error 7*8 5*3 9*5 



Corresponding to this, we have for the mean probable errors in X 

 Mean temperature 0° C. 15° C. 30° C. 



SX 3-9(£-O10- 6 X 2S(t- t')lQ-«X 4-7 (t-f) lO^X. 



The probable error is conspicuously less near the middle part of the 

 temperature range covered by the actual experiment than near either 

 limit of this range ; and this is only what we should anticipate. When 

 the mean temperature during a horizontal force observation, at Kew, 

 is 15° C, it would in the average unifilar require a difference of fully 

 10°C between the mean temperatures during the vibration and deflec- 

 tion experiments to make the probable error in q and c[ affect the fifth 

 significant figure in X. So large a temperature difference as this need 

 hardly ever be feared in a fixed observatory. 



The result is so far comforting, but does not justify the conclusion 

 that error in the temperature coefficients is a wholly improbable cause 

 of error in X. In some individual cases the probable errors found for 

 q and q' were five or six times larger than the mean. Again, in a con- 

 siderable number of instances, q and q' have been derived from only 

 one experiment. Finally it should be noticed that the probable error 

 SX increases with X, and that on the whole X is largest in equatorial 

 regions where the temperature is high, and consequently errors of given 

 magnitude in q and q' most effective. 



§ 24. The two terms i($P/r 2 ) and - |8(©/mX) in (5) were included 

 with the object of showing how errors in the values assigned to P or to the 

 torsion affect X. I have, however, no satisfactory data as to the size of the 

 probable errors in P or the torsion coefficient under normal conditions. 

 The torsion coefficient varies from thread to thread, and also with the 

 dampness of the air. It is in fact treated as variable, and is usually 



vol. lxv. 2 G 



