Hogg. — On certain Tripolar Relations. 317 



Art. XXXII. — On certain Tripolar Relations: Part 111. 

 By B. G. Hogg, M.A., F.E.A.S., Christ's College, Christchurch. 



[Read before the Philosophical Institute of Canterbury, 1st November, 1916 ; received 

 by Editors, 22nd December, 1917 ; issued separately, 24th June, 191S.] 



The equation of the circle of radius p having its centre at the point 0, 

 whose trilinear co-ordinates are (a , fi , y D ), is 



U ee aa X + b/3 Y + c 7o Z - 2ES - 2 a p 2 = o (i) 



Let d be the distance of O from H, the centre of the circle ABC. If 

 U pass through H, then 



R 2 (aa + b(i + cy ) - 2ES - 2 a d 2 = o ; 



i.e., 2ES = 2 a (E 2 - d 2 ) : 



hence U may be written 



aa Q X + b/3 Q Y + cy Z == 2a (R 2 + p 2 - d*) (h) 



If the circle U cut the circle ABC at the angle 0, then R 2 4 p 2 — d* 

 = 2Ep cos 0, whence 



U = <xa X 4 b/3 ( Y -f- cy Z — abc p cos b — o (iii) 



The equations of the circles of radius p and centre (a /? y ) touching 

 the circle ABC internally and externally are respectively 



</aX 4- b(3 Y + cy Zi — abc p = o 



aa X + bfi Y + cy Z 4- abc p = o, 



the trilinear co-ordinates of the point of contact being 



td~7T \ a o ± P cos A > A> ± P cos B > Vo ± p cos C , 



the negative sign being taken for internal contact. 



If U reduce to a point-circle, (ii) then takes the form 



«« X + bp Y 4- c 7o Z = (aa + b(3 a + c 7o ) (R 2 - d?) ; 

 i.e., aa G (X 4- & - R 2 ) + bf3 (Y + d" -■ R 4 ) 4- Cy (Z + & - R 2 ) = o. 



Let X = p, 2 , Y = p 2 2 , Z = p 3 2 , and let the radii HA, HB, HC subtend 

 at O the angles A., p., v respectively ; we then have 



aa c p! cos A. + b/3,p 2 cos p 4- cy n p : . cos v = o (iv) 



