MEAN DISTANCE OF TWO FIGURES IN A PLANE. 



731 



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

 all the points of AB from each other 



R = AB e-$ . 



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

 rectangle ABCD from the point O in its plane, and 

 POR and QOS are parallel to the sides of the 

 rectangle through O, c 



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



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

 + OP 2 . DOA + OQ 2 . AOB 



+ OR 2 . BOC + OS 2 . COD 



V 



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

 of all the points of the rectangle ABCD from each 

 other, 



. _> . . ~ 1 AB 2 . AC 1 BC 2 . AC 

 log R = log AC - g 5^2 log m - e AF 1o Sbc 



,2AB a 2BC a 25 



+ 3 BC BAb + 3 AB AUij ~ 12 • 



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



1 IT 



log R = log a + jj- log 2 + -3 - J2 



= log a — 0*8 

 R = 0-44705 a . 



25 



= loga- 0-8050866 



(7.) The geometric mean distance of a circular line of radius a, from a point in 

 its plane at a distance r from the centre, is r if the point be without the circle, 

 and a if the point be within the circle. 



(8.) The geometric mean distance of any figure from a circle which completely 

 encloses it is equal to the radius of the circle. The geometric mean distance of 

 any figure from the annular space between two concentric circles, both of which 

 completely enclose it, is R, where 



(«i 2 — « 2 2 ) (log R + g ) = a i 2 lo g a i — a 2 l°g ff 2 , 



a x being the radius of the outer circle, and a 2 that of the inner. The geometric 

 mean distance of any figure from a circle or an annular space between two con- 



