192 C. F. GAUSS OX THE GENERAL THEORY OF 



Let Falter by infinitely small but equal steps. A system of 

 surfaces will be produced, dividing space into infinitely thin 

 strata, and the inverse ratio of the thickness of the strata to the 

 intensity of the magnetic force w^ill then hold good not only for 

 different points in one and the same stratum, but also for dif- 

 ferent strata. 



8. 



We will now take into consideration the values of V on the 

 surface of the earth. 



At a point P of the earth's surface let yjr be the intensity ; 

 P M the direction of the whole magnetic force ; w the intensity, 

 and P N the direction of the force projected on the horizontal 

 plane, or P N the direction of the magnetic meridian, meaning 

 thereby the direction indicated by the north pole of the mag- 

 netic needle ; i the angle between P M and P N, or the dip ; 6, t, 

 the angles formed by the elementary portion ds oia line on the 

 sixrface of the earth and the directions P M, P N. Lastly, V 

 and V + d V correspond to the two extremities of ds. 



We have consequently 



cos 6 . = cos i cos t, (o = -yfr cos i. 

 And the equation in Art, V. becomes 



d V = CO cos t .ds 



If two points on the earth's surface P° andP', at which Fhas 

 the value of V° and V, are connected by a line traced on the 

 sm-face of the earth of which c? 5 is an indeterminate element, then 



f" 



CO COS t . ds = V — F°, 



if the integration be extended through the whole hne ; and it is 

 plain that three corollaries hold good similar to those in Ait. 

 VI., namely, 



I. That the integral / w cos t . ds keeps the same value by 

 whatever path you proceed on the surface of the earth from P° 

 toP'. 



II. The integral / o) cos t . ds throughout the whole length of 



a closed line on the surface of the earth is always = 0. 



III. In such a closed line, unless throughout its course t 

 = 90°, a part of the values of t must necessarily be acute and a 

 part obtuse. 



9. 



Propositions I. and II. of the foregoing article (which, pro- 



