o o 



included by tlie opposite spaces (straiglit lines, radii of the circle) 

 .-Tj = OP^ and .yr„ = OP^, (> being the centre of the cii;cie. 



So, in tlie case of two variables d\ and x^ with tiie quadratic form 



we have to [)ut 



^'n = 9i' ' ^32 = '],' . ^12 = '7i ?, (iosp,^, 

 whence 



E = q,' g,' sm' i>^^. 



The correlation-coefficient )\.^ now takes the value 

 ?'ij == — cos /7jj = — cos p^^ = — 



^b,,b^. 



Considering the errors in 3-dimensional space, the spherical simplex 

 is a spherical triangle P^P.^P^. 



The quadratic form H, after being transformed, reads 



Il=q,''V,'-^g.;\v^'-\-q,\v,'~{-2q.,q,,i\^.v,cosp.^^-^2q,q^.v,,v^cosp^^-\-2q^g^.v^.v,cosp^^. 



The opposite angle /J^g of the edge (or side) p,, now merely is 

 the angle P, of the triangle. Denoting, for the present, the edges 

 (or sides) bj' p^, p^, Ps, so that 



we have 



r,,= — cos P, , r, J = — cos P, , r,^ = — cos P, 

 and 



cos Pj -j- COS Pg COS Pg 



23 ' 13 ' 12 



co« P23 = COS p, = . p . „ = — .— etc., 



smP^smP^ \/^\_r^^^){\—r,^^) 



r= 1 — cos>23 — <^os>i3 — cos' p^., + 2 cos/),g cospj3 cosp^^ 



= 1 — cos' /)j — cos"^ p^ — cos^ /?g + 2 COS p^ cos p, cos Pj. 



Putting further 



/>! + P2 + P3 = 2 S , P, + P, + P3 = 2 ^, 



we may reduce T to 



r= é sin s . sin {s — p^) . sin (s — p^) . sin (s - Pf) 



— 4 cos S . cos (S—P,) . cos (5-PJ . cos (S—P,) 



sin Pj sin P^ sin P^ 

 The relation 



^ ttji ajci Bjk 



]' 



'^'^■^' 2 2E 



here involves 



