1 72 The Rev. H. Llovd on the mutual Action of permanent Magnets. 



shall be situated in the same right line. This condition is expressed by the 

 relations 



the two equations being equivalent to a single condition, inasmuch as one of 

 them is a consequence of the other. Substituting in the formulae (10, 11, 13), 

 and expanding, they become 



(^ + cos 2a) cos f + sin 2a sin f + Q ?' sin 2a = 0, (20) 



(^-cos2a)sinf+sin2acosf + ^ y'(^ - cos2a) = 0, (21) 



(^ — cos 2a) sin f + sin 2a cos f + P/ sin 2a = 0. (22) 



Dividing (20) by (21), we find, on reduction, 



cos f = 0, and therefore f = 90°. (23) 



Accordingly the plane in which the magnet c is constrained to move must be 

 perpendicular to the magnetic meridian. 



Now, making f = 90° in the three equations (20, 21, 22), the two former 

 are found, of course, to be identical ; and we have 



l-|-gj3_0j ^— cos2a-|- Pp'sin2a = 0. 



From the first of these we obtain 



which determines the place of the centre of the intermediate magnet c. Again, 

 in virtue of the relation p -\-q =i 1, there is 



Wherefore putting, for abbreviation, 



the second equation becomes (|^ — cos 2a) -f- A;sin 2a = ; and we find 



