of Terrestrial Magnetism. 355 



Now as regards the vertical force and the localization of 

 the currents, we must in the first place obtain an expression 

 for the potential V. We may take either 



\ dY v dY 



A= =— i or Y = — 



adu ' a sin udX } 



where a is the radius of the earth, and find 



Y = — a sin u cos u sin {t + \). 



This expression for V happens to be a tesseral harmonic ; 

 and the potential, if it exists, must therefore be either 



^2 



— V= — sin u cos u sin (t + X) 



or 



—Y'= -g sin ucos usm(t + X). 



For the vertical force we obtain in the first case, putting 

 r=sa, 



dY 



— -j- = sin 2u sin (t + X) ; 



and in the second case, 



dY' 3 . a . , ^ 



— -t— = -rsin zu sin (^ + X). 



Both expressions have their minima and maxima coincident 

 with those for the northerly components of horizontal force, 

 a fact which finds its confirmation in actual observation. 

 They also give us the phase of vertical force to be the same 

 for each hemisphere, and not to change as in the case of the 

 horizontal force. But there is an important distinction, 



while — -7— has its maxima and minima coincident with the 

 dr 



maxima and minima of horizontal force at latitudes greater 

 than 45°, and in the equatorial regions the maximum of hori- 

 zontal force ought to be coincident with the minimum of 



dY f 

 vertical force, and vice versa, just the opposite holds for =-• 



At Greenwich the maximum of northerly force takes place 

 at 7 p.m., the minimum at noon ; the maximum of vertical 

 force takes place at 7 p.m., the minimum at 11 a.m. 



At Bombay the maximum of northerly force takes place at 

 11 a.m., the minimum at 9 p,m. ; there is a very decided 



202 



