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



If the two points are located infinitely near together. 



V=M.^— ^ becomes V=M.- 



■ r, ., 2a . eoaO 



r^ = M . . 



r x r 2 , r~ r' 



An equivalent of the moment of the system 2aM. is as 

 4 

 before, 2aM.= -7rR,V , and the value of the potential at any 



O 



4 cos$ 

 point outside the sphere (r . 6) is V=-7tR 3 <t . . 



Typical forms of the curves may be found by employing 



simply the expression, Y= —^-. 



2 



Gauss' Theorem of Polar intercepts. 

 2. 



On the sphere whose radius 

 is R, construct a line of force 

 one of the points being (r . d). 

 Through (r. 0) draw a circle 

 with - radius r, a tangent to 

 this circle making an angle 

 ?n with the polar axis, and 

 also a tangent to the line of 

 force making the angle n 

 with the axis. The triangle 

 PTS will have the interior 

 angles I, m, n, at correspond- 

 ing vertices. Now, 



PS . cos I = OS . sin^ 

 PS . sin I = TS . cose 



tan I 



For Gauss' Theorem. 



Hence, OS . tan I = TS . cot0 = TS 



2 ' 



as can be proved by the discussion of the resolved forces. 



Therefore, 



ST= 20S, and OT = 30S. 



The intercept from the center cut off by the force tangent 

 is one-third the intercept cut off by the circle tangent. 



TJiis formula is also convenient for computing the inclina- 

 tion of the line of force to the normal as it leaves the surface 

 of the sphere. If the lines are seen on the section of the 

 meridian plane perpendicular to the line of vision, it applies 

 directly, tan 1=2 cot 6 = 2 tan m. If the angle actually seen is 

 on any other meridian section, the projected value of the 

 angle must be converted from its oblique to the perpendicular 

 plane, by turning it through the angle a. Since I is the angle 



