IM 



THE (JEOMETRICAL MEAN DISTANCE 



(2) The geometrical mean distance of P, a point in the line itself, from AB 

 U found from the equation 



AB(\ogR+ l) = PB\ogPB-PA \ogPA. 



When P lies between A and B, PA must be taken negative, but in taking the 

 logarithm of PA we regard PA as a positive numerical quantity. 



(3) If R is the geometric mean distance between two finite lines AB and 

 CD, lying in the same straight line, 



AB . CD (2 log R + 8) = AD* log AD + BC 1 log BC- A C' log A C- BD* log BD. 



(4) If AB coincides with CD, we find for the geometric mean distance 

 of all the points of AB from each other 



B Q_ 



(5) If R is the geometric mean distance of the 

 rectangle ABCD from the point in its plane, and 

 POR and QOS are parallel to the sides of the 

 rectangle through O, 



ABCD (2 log R + 3) = 20P . OQ log OA + 20Q . OR log OB 

 + 20R. OS log OC + 20S. OP log OD 

 + OP*.D6A + 0&.A6B 

 + OR* . BdC + OS 1 . COD 



(6) If R is the geometric mean of the distances 



of all the points of the rectangle ABCD from each B t 

 other, 



A&, AC .5(7, AC 



When the rectangle is a square, whose side = a, 



= log a -0-8050866, 

 R = 0-44705 a. 



