12 Prof. W. A. Norton on Terrestrial Magnetism. 



B to r it can happen that 8 will be increased and 8' diminished, 

 and therefore that sin"# sin"*', in formula (4), can remain the 

 same. Now Bs=Br cos rBs, or dS'=k cos a' ; and Bt=Br cos 



dS cos a 

 rB*, or d8=k cos a. Hence ^= , ; and, by equation (5), 



neglecting the minus sign; putting also w=angle A'BD, 



sin 8 cos d' cos a cos a 



or, 



cos 8 sin <*' cos of cos ( w — a)' 



* 



sin 8 cos <J' cos a 1 



cos <J sin 8* cos u cos a -f- sin w sin a cos w + sin ^ tan a 



sin 8 / cos # — sin 8 cos £' cos u 



Whence, tan a— : — * ^— : — — , 



; sin o cos o' sin u 7 



cot 5 tan 8 



/ 



or, tan a= = — cot u. 



7 sin 2^ 



If we put (?=ABA / w = 180-£, and 



cot 8 tan tf' 

 tana = — ^^ — -f-cot£ . . , (6.) 



This formula gives the angle DBL. Subtracting this from 90° 

 we obtain nBA, the angle included between the direction of the 

 needle and BA(8). The difference between this and ABC will 

 be the declination of the needle, which will be east or west, 

 according as one or the other of these angles is the greater. 



The first of the equations above gives the following, which 

 may be used as a tentative formula in place of equation (6) : — 



cos a tan 8 



cos {u — a) tan<J' 



(70 



To make use of formula (6) we must know <J, d\ and £ 

 These may be obtained by solving the two spherical triangles 

 ACB, ACB. The latitude and longitude of the place B, and 

 the latitudes and longitudes of the two poles A and A' being 

 given, we readily find CB, AC, and A'C, and the angles ACB, 

 A'CB. 



The formulas which have now been investigated, viz. (2), (3), 

 t and (6), serve for the determination of the vertical and horizontal 

 intensities at any place, and the declination of the needle. By 

 taking the square root of the sum of the squares of the horizontal 

 and vertical intensities, we shall have the directive force or total 

 magnetic intensity of the place ; and by dividing the vertical by 

 the horizontal intensity we shall have the tangent of the dip. 



made 



pared 



will proceed to give an exposition of the details of the calcula- 



tions 



(To be continued.) 



