344 F. H. Bigelow — Further Study of the Solar Corona. 



The Equation of the Lines of Force. 



The equation to be investigated may be derived as follows. 

 Without raising the question of the physical constitution of 

 masses of matter exhibiting the phenomena, we may under- 

 stand by a polarized sphere, one in which magnetic or electrical 

 potentials are to be referred to one general axis ; or if not sym- 

 metrical, one in which the positive potential is distributed about 

 one pole, and the negative potential about another pole, the poles 

 themselves being the intersection of two nearly opposite axes 

 with the spherical surface. The potentials at any points in 

 space may be discussed as if the actual distribution of polarized 

 matter were replaced (1) by a small magnet at the center of 

 the sphere, the direction of the axes coinciding, or (2) by a 

 surface distribution whose density varies from a maximum at 

 the poles, with the cosine of the angle of the polar distance, to 

 zero at the corresponding equatorial plane. The former is the 

 -, result of bringing two equal masses of oppo- 



site signs very near each other at the center 

 of the sphere, and the latter is the lamellar 

 distribution. They each give the same 

 equations for lines of force and equipotential 

 surfaces respectively. 



Let M x , M a be any two electrified masses 

 connected by an axis of reference ; join any 

 point P with M, and M a by straight lines, 

 and with the axis A by any curve ; across a 

 cylindrical surface generated by AP, the 

 flow of force from M t is N r and from M a is 

 ~N„ the constant total force being N" 1 +N" a =!N". 

 Let PM, be the curve formed by the locus 

 of the point that will keep N the same 

 while the position of P varies in a plane, 

 and let the tangent to this curve at the 

 point Mj make the angle 6 with the axis ; 

 N is called the order of a line of force. 

 Equation of lines of Considering a single mass, the flow of force 

 force and equipoten- corresponding to the circular zone whose 

 angle at the summit is 2#, is proportional to 

 the cap whose height is 1 — cos 6. 



47rM 



1 — COS 



N = 2ttM (1 — cosfl). 



Introducing this into the equation and limiting the case to 

 equal masses of opposite signs, we obtain, 



2zrM 1 (l— cos^ 1 ) + 27rM a (l— cos5 9 ) = 27rM(l — cos 0) = N; 

 whence, M 1 cos 6 1 + M a cos a = M a + M, cos 6. 



and by our special case, 



