xxxvi Introduction to the Makerstoun Observations, 1843. 



to the change of the vertical component of the magnetic force, which would produce 

 a disturbance = »], and hence he shews that the value of the constant tan e will be 

 given by the formula 



^ = — ^ — = tan € cot e 



Where d is the magnetic dip, T' and T the times of one vibration of the needle, the 

 former in a horizontal plane, the latter in a vertical plane. 



50. This method has been found to fail in practice ; the determination of the 

 time of vibration in a vertical plane has been proved to be a matter of much diffi- 

 culty, if at all possible, mixed up, as it is, with several sources of error, which are 

 not easily accounted for or eliminated.* It has been shewn with respect to the 

 time of vibration in a vertical plane, 



* The following statical method might be substituted for that of Dr Lloyd : — Let ri be the small 

 angle which the magnetic axis of the balance needle makes with the horizontal, then the equation of 

 equilibrium is (48) 



m Y cos ;j = W (/ cos (e + »)) (1.) 



if a magnet, whose moment is M, be placed vertically, with its centre at a distance r from the centre of 

 the balance needle, and in the continuation of the magnetic axis of the balance needle when horizontal, 

 the needle will then make an angle d wi.h the horizontal, and the equation of equilibrium will be 



Y'cosa = W^r cos(e + 5) (2.) 



m 



where 



Y' = YT^ (3.) 



From equations (1), (2), and (3), 



c M tan d — tan jj . . 



■i * Y ~ cot € — tail jj ^ '^ 



Now, if X be the horizontal intensity of the earth's magnetism, and d be the magnetic dip, 



Y = X tan ^ ; (5.) 



and if the deflecting magnet be placed in the line at right angles to the magnetic meridian passing through 



the centre of a freely suspended magnet, and u be the angle of deflection when the centres of the magnets 



are at the distance r^, then (No. 26.) 



cM - r,3 

 ^;^ = i^tan« (6.) 



By equations (4), (5), and (6), 



-^ tan u -\- tan u 



tan € = — ;; r ; ; = r — r, approximately : 



2 tan 6 (tan o — tan jj) — tan u tan n 2 tan d tan {d — r]) 



and if 9) = 0, or be very small, as it is in general, when the needle may be considered horizontal, then 



_ r^^ tan u .„ . 



tan e „ ., 



2 r^ tan tan d 



If the deflecting magnet be placed at right angles to the suspended magnet (as in Dr Lamont's 

 method), then sin u must be substituted for tan u. 



The 



