154 



(Pi, T\ ...Pj-i, Pj+i ,... /\) and {I\, P., .'. . P;t-i , Pk+i ,... P,)]. 

 l>iiil(liiig- ilie pósitive-deliiiite detenuinaiit 



1 , cos pi2 , cos pi3 , 



cos pi> , 1 , cos pi^ , 



cos JC»13 , cos p 23 , 1 • , 



cos pi: 



cos p->p 



cos p3p 

 1 



cos pic , cos p2p , cos p3c , • • • 



and representing by Cjk the minor of cos pji^, we have !)}• the theory 

 of the spherical simplexes 



Cjk 



cos Tijk = — 



\^Cii Ckk 



Snbstituting 



h.i = Q/ ' hk = Q.i If^ '^^^' P.i^ 

 the quadratic form H in the expression for the [)rol>abüity W trans- 



fOilTiS to 



II=:E bjj xf + 2 2" hjk xj xk — 2 {qj xjY -^2 2: cos pjk {gj xj) {qk xk). 

 This form is positive-definitive, when 



r>o, 



or, in other words: ivhen the arcs pjk are the edges of a Q-dimensional 

 spherical simplex. 

 Furthermore 



and 



whence 



E=nqr . r 



1 



F 

 Tlq^ 



Bjk — X Cjh 



9j qk 



B 



m = 



'Jk 



Cjk 



=r — cos Tljk. 



VBjjBkk ^CjjCkk 

 So, putting H in the form 



^ = - (?i ^if + ^ ^ (^os Pjk {qj Xj) {qk Xk), 

 the arcs pjk must be the edges of a ^-dimensional simplex and 

 moreover: the coirelatioii-coefficients are, hut for the sign, equal to 

 the cosines of the "^opposite angles'' Ujk. 



In the case of "errors in a plane" only a circle-biangle P^P^ is 

 to be considered. Then the arc P^P^ = p^2 equals the angle U^^ 



