236 



MR. DAVIES'S GEOMETRICAL INVESTIGATIONS 



assistance which it is in the power of men of science to afford will be freely offered 

 me ; and especially in furnishing such observations as they themselves place most 

 reliance on, together with the circumstances under which the observations were made. 



XIV. — On the Magnetic Curve. 



Let F^ and F^^ denote the intensities of the forces 

 situated in the two poles T, U ; and /3 the angle 

 which the needle, subjected to the action of those 

 forces and situated in a given point N {xy), would 

 make with the axis of (x) the magnet itself. Let also 

 r 2 = 3/2 _j_ (^ + af 



r,^=y'^-\- {oc - af. 

 Then the usual considerations give us 



F,(;t- + «) .). F,(. 



f7J^ = »«»'^ (37.) 



But if we represent tan /3 by ^, we shall have the differential equation of the curve 



to whicii the needle will be a tangent, and which passes through that point, xy. To 

 find the equation of the curve itself it only remains then to integrate 



rf r^f 



_ ^y 





dx 



(38.) 



u 'It 



Multiply the numerators of all the terms of the first side by y, and also multiply 

 out the denominators ; then there will result 



F,fdx- F,2/{a; + a)dy Y^^y^dx - F„y{x - a)dy 



In the former of these numerators add and subtract F^ (x + a)^ dx, and in the latter 

 F,^ {x — ay dx\ then we shall obtain another form, which is immediately integrable. 



It is 



F/ {y + (^ + of} dx - F^ (^ + a) {y dy ^ [x^ dfdx} 



rf 



I Fy<{y + (^ — of} dx — F;,(:r — a){ydy + {x — fl)<^^} _n. 



"T" r^ ^» 



the integral of which is 



F/(^ + «) , F;,(.r-g) __c_ 



r^ I" Tt, ""a' ^^^-^ 



in which — is the arbitrary constant which particularizes the individual curve we 



