CONCERNING TERRESTRIAL MAGNETISM. 



245 



The state of the physical problem is not at present such as to render any further 

 mathematical details respecting this curve necessary in this place. 



XVIII. — On the Points of the Earth's Surface at which the Needle takes a position 



vertical to the Horizon. 



As our hypothesis is that of two poles, or resultant centres of force, the freely- 

 suspended magnetic needle will always lie in the plane which passes through its own 

 centre and the centres of magnetic force. The dipping-needle lies, therefore, wholly 

 in the plane passing through the place of observation and the true magnetic poles. 

 But when the needle is vertical to the horizon, it passes through the centre of the 

 earth ; and hence the plane of the magnetic meridian also passes through the centre, 

 and makes with the sphere a section, which is a great circle. Also, as this plane then 

 passes through three points not in a right line, it is unique ; or, in other words, there 

 is only one circle of the sphere in which the needle can be placed to be capable of 

 taking a vertical direction, and that is a great circle. 



It is also obvious, from the expressions already given for the inclination of the tan- 

 gent to the radiants from the poles to points in the magnetic curve, that there are 

 only isolated points in that circumference in which the phenomenon of verticity can 

 take place ; and it is our business in this section to inquire into their possible num- 

 ber, and the method of determining their actual number and their respective po- 

 sitions. 



XIX. — Let O be the centre of the 

 great circle in which we have just 

 shown all the possible vertical nee- 

 dles must lie, and T U the magnetic 

 poles, and N one of these points. 

 Take the centre of the circle as ori- 

 gin of coordinates. 



Let T U be denoted by the coor- 

 dinates a^ h^ and a^^ b^ respectively ; 

 then the equation of the magnetic 

 axis is 



{x-a){b„^b)=:{y-b;)ia„-a)i66.) 

 Also the equation of O N is 



^y-^'i/ = 0, . . . (67.) 

 where jc'y' are the coordinates of N. 



The intersection of these gives the coordinates of R, the point where the tangent to 

 the magnetic curve at N, and whose poles are T and U, intersects the magnetic axis. 



From (66.) we have 



