CHAPTER II. . 

 MATHEMATICAL INVESTIGATIONS. PRELIMINARY RESUME. 



91. The calculation of the Field of Force for the assumed polar current-system. While 

 studying polar perturbations of the most varied character, we have constantly met with what we called 

 the typical field for an elementary polar storm. We have also indicated the kinds of current-systems 

 that might be naturally supposed to give rise to such fields. In Art. 36 we moreover worked out 

 a little calculation in order to obtain some idea of the distribution of intensity in this field of force. 

 We there selected the simplest possible form of current-system, namely a linear current consisting of 

 two vertical portions, which were connected with a third portion that was parallel with the tangent to 

 the principal axis in the storm-centre of the current-system. 



Our only aim in the earlier calculation was to prove the reversal in the direction of the force 

 which took place in the point of convergence, or that of divergence, when one moved from the storm- 

 centre out along the transverse axis of the system, and to obtain some idea of the proportion between 

 the magnitudes of the forces in the storm-centre and at great distances. 



A more complete calculation of the field of force for such a system might, however, be of some 

 importance, and we will therefore make one here. 



During great perturbations, the area of precipitation, as we have frequently pointed out, will extend 

 over large parts of the auroral zone, thus causing the principal axis, or those districts in which the most 

 powerful forces occur, to assume approximately the form of parts of a small circle. Very often, indeed, we 

 find conditions which indicate the existence of an entire current-circle. Instead, therefore, of the current- 

 system previously employed, it would be better to use one in which the rectilinear horizontal portion of the 

 current is replaced by a curved portion. The actual calculation will thereby be made a little more 

 complicated; but, as we shall see, a considerable advantage will be gained in another way. 



We will consider, then, the effect upon the earth of a current-system consisting of two vertical 

 rectilinear pieces of current, in one of which the current, from infinity, will approach the earth as far 

 as a height //, and in the other continue, from the height //, out into infinity, the two pieces being 

 connected by a curved piece of current lying at a constant height /; above one particular small circle, 

 whose spherical radius is C. 



We do not, of course, mean that the separate active corpuscular rays, which we assume to be the 

 cause of the storms, move in accordance with a diagrammatic arrangement such as this; the whole thing 

 is only an endeavour to find out how near we can get to the true perturbation-conditions, if we assume 

 that the integral effect of all the rays in a system of precipitation is replaced by a linear current- 

 system of this form. 



We will first look at the effects of the vertical currents. 



As our system of coordinates, we will employ a rectangular Cartesian system, with its origo in the 

 centre of the earth. We will further take the axis Z perpendicular to the plane of the current-arc. 



As polar coordinates we will employ the signs Q, 6 and w, being the distance from the origo, 

 6 the angle formed by the radius vector and the positive axis Z, and w the angle between the plane 

 A'Z and the plane through the axis Z and the radius vector. 



