162 The Rev. H. Lloyd on the mutual Action of permanent Magnets. 



and denoting the total force by R, and the angle which it makes with the axis 

 of abscissae by w, 



tan«. = -jl, i?= -^ \ (4) 



Now, stopping at the first dimension of - m the development of the trino- 

 mial factor, 



(l -2%os0 + -^) ^ = l+3^cos0, q.p. 



and substituting, we find 



^, =;= 2 COS \ qrdr = 2 Mcos 0, B^ = sin V qrdr = Msin <f> ; 



putting, for abbreviation, 



\ qrdr = M. 



Finally, substituting these values in (3) and (4) 



^ iMm , ^ Mm . , ... 



X = ^— cos 0, Y = — ,- sm ; (5) 



a^ a'' ^ ' 



j\f iffi 



tan 10 = -|- tan 0, R = — j v 1 -{- 3 cos^ 0. (6) 



The theorems expressed by the formulae (6) were taken by Biot as the basis of 

 his well-known theory of terrestrial magnetism. 



If we desire to push the approximation further, we must include (in the 



r^ 

 development of the trinomial factor) the terms involving -y. We thus find 



Aj = 2Mj cos (5 cos^0 — 3), B^ — ^M^ sin (5 cos' - 1) ; 



in which we have made, for abridgment, 



V qr'drzziM.^. 



