240 



MR. DAVIES'S GEOMETRICAL INVESTIGATIONS 



as there will be such points in each of the consecutive meridional sections, there will 

 be a series of consecutive points on the surface of the earth at which the needle will 

 be horizontal. These points lie in a continuous curve, which has been called the mag- 

 netic equator ; and we proceed to inquire into the character of that curve. 



XVI. — On the Magnetic Equator. 



Let T U, as before, be the poles 

 of the terrestrial magnet, and TNU 

 that one of the magnetic curves 

 which touches the corresponding 

 magnetic meridian of the earth 

 QNRinN. 



Put for the moment the equation 

 of the circle Q N R under the gene- 

 ral form 



{x - cf + (y - hf = r2 ; 



(50.) 



and denoting by ^^0 and g^,0 the 

 coordinates of Q and R respectively, 

 we shall find 



c = €^', andr2 = i2 + (fi^'y (51.) 



Inserting (51.) in (50.), and reducing the equation, the circle is finally expressed by 



^[-gi + g,^-\-g^S,i^y'' - 2 % = . (52.) 



This involves the arbitrary quantity b, which is the parameter upon which the iden- 

 tical circle depends. 



d y 



Since this circle is to touch the magnetic curve at some point N, the values of -^ 



derived from the equations of the circle and magnetic curve at their common point 

 {xy) must be equal ; and as the arbitrary constant in the equation of the magnetic 

 curve vanishes by differentiation, we shall have three equations between which to 



eliminate the indeterminate quantities -^ and h. This elimination will leave one 



equation between w and y, which will designate the locus of the point N. 

 By differentiating (52.) we obtain 



dy_ _ 

 dx 



(53.) 



And the differential equation of the magnetic curve is, from (38.) and F^ -|- F^^ = 0, 



^y 



dx 



y y 



3 



x-'ra x—a' 



(54.) 



'H 



