TERRESTRIAL MAGNETISM. 201 



manner accordina; to u, — ^^ — ^ — o ; V + R T having a 



value independent of u, or, what is the same thing, constant in 

 all the points of a meridian, — it must hence also be absolutely 

 constant, because all meridians converge and meet at the poles. 

 If we call the value of V at the north pole = F*, then 

 V*-V 



T = 



R ' 



dT 



and hence Y= . 



sinM . d\ 



This result may also be expressed as follows : 



-. 1 r^" dx . 



X = —. / -r^.du. 



smWt/ c 



d\ 



16. 



This remarkable proposition, that, if the component of the ho- 

 rizontal magnetic force directed towards the north be given for 

 the whole surface of the earth, then the component directed towards 

 the west {or towards the east) follows of itself, is true, conversely, 

 only with a certain modification. If Y be expressed by a given 

 function of u and \, and if t/^ represent the indeterminate integral 



/ sin u . Y d\, u being assumed constant in the integration, 



then — — j^ ' = 0, or V+R C/has a value independent of X, 



and is, generally speaking, a function of u. Thus — ^ — ^-^ - 



= ^— — X is such a function ; that is to say, the formula ^ — 

 du •' du 



gives an imperfect expression for X, a part of it containing 

 u only remaining undetermined. This want would be sup- 

 plied if, besides the expression for Y, we had also that for X, 

 for some one given meridian, or to speak generally, for some 

 line extending from the north to the south pole. We see there- 

 fore that, if we know the component of the horizontal magnetic 

 force in the direction towards the west for the whole of the 

 earth's surface, and the component in the direction towards the 

 north for all points of some one line extending from the north 

 pole to the south pole, the latter component, for the whole of the 

 earth's surface, follows of itself. 



