394 



Prof. A. Gray on the Attraction of 



the property of pairs of corresponding points. Also/!, g u A t 

 =faja', ghjb' , hc/c' and x\ y\ z'=xa'/a. yb'/b, zc'/c, where 

 a', A', c' = \/.a 2 -t tt, \fb 2 + u, sj&-\-u. 



Efe. 1. 



j\ T ow let y^,^/, be the lengths of the perpendicular from the 

 centre on the tangent plane at P, and the tangent plane at A' 

 respectively, and 0' denote the angle between the latter per- 

 pendicular drawn outwards and the line PxA'. The lemma 

 to be established is expressed by the equation 



p' cos =2> cos 6' '. 



In order to express cos O we have the direction cosines of 

 QP and AP. These are {//(a 2 + it) , g/(b 2 + u) , h/(c 2 + u)}pjh 

 and (/— ,r, <7 — //, h—z)]r. Hence 



cos 0„= f 5 (-4 / -'') = & ( k -X-gr-X • (8) 



k \a? + u r J rk\ a 2 -Y u ) v J 



or as we may write it 



cob *,=•&(* -2^) (8') 



Similarly we obtain 



cob^ = ^/*-24^-^ .... (9) 

 rk\ a 2 + u J 



so that 



p' cosB o —p o cos0 r , (10) 



which was to be proved. A similar relation of course holds 

 also for the points P a , A on the homceoid. 



