CONCERNING TERRESTRIAL MAGNETISM. 247 



The determination of .r and y from these two equations will require the solution of 

 an equation of the tenth degree. For putting (76.) under the form 



(*/ ^ - f^iV? W + V + ^2 _|. ^2 _ 2 («^^ X + h^^yf 

 = (*/i ^ - «/;y) W + ^,2 4. ^2 + ^2 _ 2 [a^ X + h^y)'Yy 



we have it converted at once, by means of {77-), into 



{b^ x-a,yY [V 4. b2 + r2 - 2 (a, a: + A, y)^ 1 . 



= (i,x-a,3/)h2 + ^^2 4^2_2(a,a. + ^.3/)P. J ^ '^ 



Hence by means of (77-), which is of the second, and (78.), which is of the fifth, 

 degree, we obtain an equation of the tenth, from which to determine x or y, and hence 

 to find the points at which the needle will be vertical. Still the reduction is ex- 

 tremely laborious, and hence, also, our means of determining how many of the roots 

 are real, and how many are imaginary ; that is to say, how many of its roots are 

 compatible, simultaneously or separately, with the values to which a^ b^ and a^ b,, are 

 limited by the physical which must be appended to the algebraical conditions from 

 which the equations (77-) and (78.) were formed. Those conditions are r^ > aj^ + b^, 

 and r^ > a^j^ + b^j^, that is, of the poles being within the earth ; but whether the per- 

 pendicular from the centre of the earth upon the magnetic axis intersects that axis 

 between the poles or not, cannot be, a priori, nor yet from any knowledge furnished 

 by experiment, at present determined. As, however, all the cases that can arise 

 from all possible positions of the two poles are included in the above formulae, and 

 as they evidently cannot simultaneously exist, we are entitled to infer that all the 

 roots are not simultaneously real. In the absence, however, of these considerations, 

 we learn that there are not more than ten points on the surface of the earth at which 

 the needle can be vertical ; and that whatever may be the number of them, it is at 

 all events even, viz. 2, 4, 6, 8, or 10. We also learn that how many soever of these 

 be real, they are all in one plane passing through the centre of the earth ; and with 

 respect to the great circle in which it cuts the terrestrial surface, taken as the axis 

 of spherical coordinates, all magnetic phenomena on the surface of the earth are sym- 

 metrically disposed. How far this is verified, within the limits of errors of observation 

 and of local interference with the full development of the effects of the magnetic 

 force, has not yet been inquired into. Indeed, till this plane has been determined it 

 would be impossible to conduct the inquiry in a direct manner; and as only one of 

 these points is yet actually assigned with any close degree of approximation, we 

 are not yet in a condition to enter upon the inquiry. Still, these facts combined with 

 observations relative to other phenomena, especially respecting dip and intensity, 

 (the variation for obvious geometrical reasons included,) accurately made, might 

 furnish important aid in a tentative determination of the plane itself; the method of 

 proceeding in which must be sufficiently obvious to those inquirers to whose minds 

 the geometry of coordinates is familiar. 



