SOLUTION OF THE EQUATIONS. 613 



Taking ['0014542] to represent the number of which '0014542 is the 

 logarithm, we have 17 = ['0043626], F 2 ' =['0058168], &c. as in the following 

 equations, and Z 1 = 2F 1 1 = ['3053926], &c. 



Also 



Z - -1 Z - 4 r 7 -- ( V Z - -*** 



3 7 ' 5 7 35*' 



z _40 4 _32 , _ _112 g _ 1024 7 



'-. a 68 r-, ^_ 6 - 77 ?, ^_ 7 429 r, ^_ 8 - 6435 ^, 



8 7 = 

 12155 46189 



and Y_ n l = - X_, L l = Y n l x r-' l+ \ Therefore F., 1 = 1 = - Z., . 



Let the magnetic forces at the North pole, considered as a point in 

 the zero meridian, be X a, Y=b, Z=c, where a is positive when 

 directed towards the north in that meridian, b is positive when in the 

 direction perpendicular to that meridian and towards the west, and c is 

 positive when directed downwards. 



Then if, instead of as above, we consider the pole to belong to the 

 meridian whose east longitude is X, we shall have 



X = a cos X + b sin X, 

 Y a sin X + 6 cos X. 



Similarly let a', b', c' denote the magnetic 

 forces at the South pole. 



Then we shall have, expressing X, Y, Z in 

 terms of the Gaussian constants, 



-[ '0043626]^' -[ '0058168]^ -[9-9103610> 3 1 -[9'7656872]^ 4 1 -[9'591050l] i / 5 1 



- [9-3962097] gr. 1 -[9-1868105] g, 1 -[S'9664160]^ 1 -[87374212]gr 9 1 -[8'5015145]^ I0 1 = a, 



-[ '0043626] A, 1 -[ 'OOSSieS]/^ 1 -[9-9103610]^ -[9'7656872]/i 4 1 -[9'591050l]/( 5 1 



- [9-3962097] V- [9-1868105] V-[8'9664160]A 8 1 -[8-7374212]V-[8-5015145]A 10 1 = 6, 



and 



