CONCERNING TERRESTRIAL MAGNETISM. 237 



may have occasion to consider. It is determinable from any one condition; as 

 passing through a given point, touching a given line, &c. 



But — - — and — - — are the cosines of the angles which the directions of the com- 



ponent forces make with the axis of x^ that is, with the magnetic axis; and hence, de- 

 noting these by ^^, &^^, (being estimated from the same branch of the axis,) we shall 

 have 



F^ cos &^ + F^^ cos ^^^ = 2 cos |3', (40.) 



where /B' = cos" ^ -x—. 

 ' 2 a 



This equation having the values of r^, r^^ and )3 restored, becomes 

 F/ (-y + a) , F,^ {x - a) _ c 



>/y^ + (^ + «)3 "^ -/y + (^- - a)2 2a^ C41-; 



which, when deprived of its radicals and denominators, is of the eighth degree. 



We might, however, obtain this property in a different manner ; and as it also fa- 

 cilitates the investigation of one or two other theorems which we shall require here- 

 after, it may with propriety be added here. 



Let T and U be the poles, N the centre of 

 the needle. Let, as before, the forces be F, 

 and F/, ; let T N S and U N S, the angles made 

 by each of the component forces and the re- 

 sultant one, be called A^ and A^^ respectively. 

 Let r^ and r^^ be the distances T N, N U, and 

 ^^, &^^ the angles N T S, N U S ; and take N h ; 



F F 

 N ^ : : -| : -f, and complete the parallelogram 'S kng. Then N n is the position 



of the needle. Produce it to meet the axis T U in S ; and draw the perpendiculars 

 T K and U L upon N S. Then by the composition of forces 



In 

 sin 4 p_ sinj ?Nw "| _ N;j _ J^ __ F/, r/ 



sin Jy 



Denote now the angle N S T by 2, and TS, U S by # and u respectively. Then 



t 



i/yL~singNwJ "Ny" F, "" Y^'r,f ^^ '* 



sm JS" 

 sin A^ r^ t Vi, 



^^i'^if-sin^-""^ 



r,, 



(43.) 



or by comparing (42.) and (43.), we obtain, 



— — ?m!± ...... ^44 ) 



a relation which, when F^ ± F/, = 0, is already known*. 



* Vide Leslie's Geometrical Analysis, art. " Magnetic Curves" ; or Mr. Baklow's Treatise on Magnetism 

 in the Encyclopaedia Metropolitana, p. 794. 



