MATHEMATICAL COMPONENTS OF DISTURBANCE. 



173 



of terrestrial magnetism are a little suggestive of the existence of such fields, and it is well to bear the 

 possibility of their existence in mind. 



89. The elements which are usually recorded by magnetographs are the Vertical Force, the Horizontal 

 Force, and the Declination. By an increase in Vertical Force is meant a force AV tending to pull towards 

 the Earth's centre the dipping pole of the magnet. A + sign to AV thus denotes a force urging the 

 N-pole of a magnet towards the Earth's centre when the N-pole is, as at Kew, Falmouth, and Colaba, 

 a dipping pole. But at Mauritius, Christchurch, and Winter Quarters, a + sign attached to AV means 

 a force urging the N-pole from the Earth's centre. An increase AH in the Horizontal Force means a force 

 urging the N-pole in the direction of the magnetic Meridian drawn towards the magnetic Pole in the 

 northern hemisphere. An increase AD in the Declination means a force II AD urging the N-pole 

 perpendicular to the magnetic Meridian, in the direction of D increasing. At Kew AH when of + sign 

 means a force on a N-pole inclined at 16|" to the west of geographical north, while AD when of 

 + sign means a force on a N-pole (tending to increase westerly Declination) inclined at 16J" to the 

 south of geographical west. At Colaba, where the Declination is easterly but nearly zero, AD when + 

 means a force acting nearly due east. At Christchurch, where Declination is about 16J east, AD when 

 positive means a force HAD inclined at about 16^ to the south of geographical east. 



Thus, even as related to the local geographical directions, increments in D, H, and V have at the 

 different stations widely different significance. As referred to fixed axes in space, the significance is even 

 more complex. Christchurch and Falmouth, for instance, differ nearly 180 in longitude, so that what is 

 east at the one is west at the other. 



The above considerations will show that in studying disturbances it is desirable not to confine our 

 attention exclusively to the disturbances AD, AH, and AV, but to take account also of the disturbances 

 AN and AE to geographical north and east, and, finally, to regard the disturbance at any place as a vector 

 possessed of magnitude and having directions referred to three fixed axes at the Earth's centre. 



For co-ordinate axes let us take the Earth's axis as axis of z, a perpendicular axis in the Meridian of 

 Greenwich as that of y, and a second perpendicular axis in the Meridian 90 east of Greenwich as axis of x. 

 Let AD, AH, AV denote the disturbing forces experienced at a place of latitude A. and (easterly) longitude I, 

 assuming AD to be counted positively when easterly Declination increases, and AV to be counted positively 

 when it urges a N-pole towards the Earth's centre. (We are here departing, it should be noticed, 

 somewhat from the common usage.) Then for the corresponding components AX, AY, AZ relative to the 

 system of co-ordinates specified above, I find 



AX = HAD (cos I cos D + sin I sin D sin X) + AH (cos I sin D - sin I cos D sin X) - sin I cos X AV, 



AY = H AD ( - sin I cos D + cos I sin D sin X) + AH ( - sin I sin D - cos / cos D sin X) - cos I cos X AV, } (8). 



AZ = - HAD cos X sin D + AH cos X cos D - AV sin X 



The geographical co-ordinates of the several stations and the mean value of D for the epoch 1902-3 are 

 given in the following table : 



In the formula, D is to be regarded as positive when Declination is easterly as at Christchurch, and X 

 as positive when latitude is northerly as at Kew. Where the S-pole dips, AV is to be regarded as 

 negative when the numerical value of V increases. 



