98 Prof. Miller on the Anharmonic Ratio of Radii 



Hence siu PQ sin SR th + ik + gl pw + q?^ + nv 



sin QR sin PS " " pA + <\k + vl eic + fo + gw ' 



(3) 



(4) 



where the lower sign is to be taken when one only of the points 



Q, S lies between P and R. 



Suppose PQ, PR, PS to be all measured in the same direction 



from P, then 



sinPQ sin (PR — PS) _ di + ik + gl \>u + ({v + yw 

 sinQR sin PS ~~pA + qA + r/ ew + fv + gw 



Also, sin QR = sin PR sin PQ (cot PQ - cot PR), 



sin SR= sin PR sin PS (cot PS - cot PR) . 

 Hence 



^^ + ^^^^^ (cot PS - cot PR) = 4^tf^(cotPQ-cotPR). (5) 

 pw + qu + Yw ^ ^ p/i + qA; + v^ 



When PQ, PR, and the symbols of P, Q, R, S are known, 

 PS may be found either from equation (5), or from the equations 

 /)_ eA + fA + g/ pw + qv + rw sin (PR — PQ) 

 ~ pA + qA 4- r/ CM + fi' + g«<^ sin PQ ' 



tan(PS-iPR) =taniPRtan (^ -^)- 



When PQ, PR, PS, and the symbols of P, Q, R are known, 

 the indices of S may be found by means of the equation 

 pw + qw + rw _m 

 CM + fy + gw n ' 

 m, n being integers, where 



m _p/^ + qA: + rZ sinPQ sin SR 

 n ~ e/i + fA + g/ sin QR sin PS 

 _ pA + qA: + vl cot PS — cot PR 

 "eA + f/t + g/ cot PQ- cot PR' 

 and the equation 



UM + TO + viw = 0, 



where uvw is the symbol of the zone-circle PR, and therefore 



^=fr-9q, v=gp—er, m = eq—fp. 



Fig. 1. 

 Let be the centre of the 



zone-circle PR. Draw qr 



parallel to PO, meeting the 



straight lines PQ, PR, PS in 



q, r, s. Then 



sr 

 qr 



cot PS - cot PR 

 cot PQ- cot PR" 



