INCLINATION. 595 



We shall again assume that the angles Ij and I 2 differ very little 

 from each other, without which the needle would be too defective. 

 The angles I a - l l and I 2 - I a are then very small, and we may 

 accordingly replace their sines by the corresponding angles in the 

 ratio of the equations (17) and (18); we deduce from this 



l a -I l M 2 cos 



I 2 -I a M lC osI 2 ' 



The second member of this equation differs little from unity; 

 for the angles l l and I 2 are supposed to be very near, and the 

 magnetisations also as equal as possible. It follows, to within an 

 error of the second order, that 



(20) 



The angle I' is really given by the mean ot two readings for 

 the two points of the needle, and of two other readings after the 

 frame is turned through 180. The angles I x and I 2 are then each 

 the mean of eight readings. The value I a which is to be observed 

 is given, without appreciable error, by the mean of sixteen readings 

 corresponding to reversals and to change of magnetisation. 



It will be seen that, in the same conditions, the want of homo- 

 geneity of the agates is eliminated by a second observation made 

 by turning the frame through 180. 



If the magnetisations were very different, the ratio m of the 

 moments Mj and M 2 could . be determined by the times of oscil- 

 lation in the same azimuth. Equation (9) will then give 



i+m 



It is ascertained, in all cases, if the times of oscillation are 

 materially different; but the introduction of this new correction, 

 if it were necessary, would be a disadvantageous condition. 



When the reversal of the needle, the inverse magnetisation, 

 and the rotation of the frame, give variations of several degrees 

 in the reading, the apparatus is manifestly imperfect, and no mode 

 of correction will find an exact inclination. 



QQ2 



