Il8 COEFFICIENTS F INDUCTION. 



s being the distance of the element ds from the base of the per- 

 pendicular upon the line. From this we deduce 



If the perpendicular x falls at the end of the line, we have S Q = 0, 

 and therefore 



/ . R = / . *Ja 2 + x 2 - i + - arc tan - . 



2nd. Multiplying this expression by adx, we shall have the geo- 

 metrical mean distance from a point to the rectangle adx, which will 

 enable us to calculate the values of R relative to the rectangle. 



We get in this way for the summit of a rectangle ab, 



1-3 ^ a b b a 



2/ . Kfl + 3 = 2/ . v<* 4- + -7 arc tan - + - arc tan - . 

 b a a b 



If the point has any given position, we may decompose the 

 rectangle into four others having a summit at this point. 



3rd. The action of a uniform current which traverses a cylinder 

 whose section is a hollow cylinder, being zero in the interior, and 

 inversely as the distance from the axis for an external point, it follows 

 (757) that : 



The geometrical mean distance of a point to a circumference is 

 constant ; it is equal to the radius if the point is on the inside, or to 

 the distance from the centre if the point is on the outside. 



4th. The geometrical mean distance from a point to a corona 

 bounded by the radii a^ and a is equal to the distance from the 

 centre if the point is on the outside. 



If the point is inside the corona, the value of R is the same as 

 for the centre, which gives 



. R = V rdrl . r = 27r 



Jk 



a 2 / . a a\l . a\ 



R = 



5th. The geometrical mean distance from a corona to any given 

 figure is equal to the geometrical mean distance from the centre to 

 the figure if the latter is entirely without the corona. 



