PART I. ON MAGNETIC STORMS. CHAP. IV. 



By inserting this value for h in (7) we obtain 



J) 



and 



tan 



h i i/ i d* 

 q>, = = ~ I/- - , 

 a, <J " x i 



37 



(9) 



By the above equations, // and / are determined as functions of d. On account of the direction 

 of the vertical components, we have 



d>o. 



If our assumptions are correct, we must have real quantities, and the strength of the current 

 must be finite. We then obtain 



<5<Cy -j where ]/ -- = 0.402 . 

 ' x ' /. 



It is easily ascertained that the function for // in this interval has neither maximum nor minimum. 

 As the function in the interval considered is continuous and finite, we may conclude that it has its 

 extreme values at the limits of the interval, and especially in such a way that we get the greatest 

 height when d = o. 



In the case of / we find that the function has a minimum for the value d = - that is to say 



x 



for a value within the interval considered. 



Still narrower limits may be set to the interval, however, if we now make use of our knowledge 

 of the vertical intensity at Axeleen. 



The sensibility of the balance was determined, after the return of the expedition, as 24.6. If, 

 therefore, we employ a value of 35, there is no doubt that it is too high. We then obtain 



tan 



P " 

 == -^77 > 0.885 < 



or 6 <d 0.312 . 



o and 0.312 can thus be employed as the limits for <J. 



In the following table, the height and strength of the current are calculated for 4 values of <5, 

 namely d = o, - , 0.263 an d 0.312. The value d = 0.263 anwers to a sensibility of the balance of 



X 



24.6, and therefore the values we obtain there should be the nearest to the true values. 



TABLE XLIII. 



We see that even if we pay no attention at all to the vertical intensity for Axeleen, we may still 

 conclude that the current cannot lie higher than 368 km., answering to a current-strength of 342,000 

 amperes, and also that the current-strength cannot be less than 314,000 amperes, provided our assump- 

 tions in other respects hold good. 



Considering that P, for Axeleen is known with very fair accuracy, the true values should lie near 

 those that answer to d = 0.263. 



The values found for h and i are, as we shall presently see, comparatively small in this pertur- 

 bation, indicating that the perturbation is comparatively slight, and of rather a local character in the north. 



