THE THEORY OF TERRESTRIAL MAGNETISM. 441 



At a point where two branches of the line cross each other this 

 expression must have two values, hence at such a point 



, r dY ^dX , v dY ^ 



X -f-- - F -jr- = and X -j- Q - 1 - Iff = : 

 rfX rtX d0 (W 



these are the two equations for finding the values of X and 6. 



At points of maximum or minimum declination, the same two equations 

 must hold good. 



The difference between this case and the former is that in the case 

 of maximum declination 



d dY dX\ t , d ( Y dY_ v dX 



2 77- I clUQ ~~T~/\ I -A- ~~-n\ J~ 



<M dx d\) 1 d0\" do de 



must both be negative, and in the case of minimum declination they 

 must both be positive, but in the case of two branches crossing each 

 other they must have opposite signs. 



Proceeding to a second differentiation we have at such points 



d'Y v d>X\ I # Y #X \ 9 * a ^/ Y d 2 Y d*X\ 



X r^r, * ia , (ov) + X j2i"j\ ~ * JQ j\ 2ovb\ + X -j; Y -~j~, (b\)- = 0, 

 dP d0- / v \ dOd\ ddd\) \ d\- dX ] v 



which will give the two values of , at such points. 



ci\ 



At points where the horizontal force is a maximum or a minimum 

 we have 



X-+ Y- a maximum or a minimum, 



hence the values of 9 and X for such points are given by the equations 

 . dX -, r dY , dX ^, d Y 



x + Y = 



similarly the relation between 80 and SX for the tangent line to the line 

 of equal horizontal force is given by the equation 



dX dY\ (dX dY 



Suppose V to be the magnetic potential and to be a function of 

 and X. 



A. II. 56 



