413 



and the sines of half the transverse common tangents by ni% n% we 

 get from the theorem of Art. 25, and from equation (45), the equations 

 of the four pairs of circles which touch three small circles on the sur- 

 face of a sphere, as follows : — 



I IS j mS' I nS" _ Q < 



\cos r \cosr' *cosr'' ' 



FW tnfW_ I n'S" = . 



'Vcosr Vcos r' \cos r" ' 



I TS + m&_ + J n S" = Q _ 

 \cos r \cosr' \cos r" 



VW \ rnfW I 

 \ cos r \ cos r' A/' 



COS f 



(46) 

 (47) 

 (48) 

 (49) 



31. The four equations, (46)-(49), when expanded, are equivalent 

 to the following determinants : — 



o, -L, 



cos r 

 ' n 



m 

 cos r" 



cos r' 





cos r"' 



m 







cos r' 



? u ' 



cos r 





<S', 



8" 



0. (50) 



cos r cos 



, S, ! 



cos 



S 0, 



cos r 



0, 



cos r cos r 

 5, ^, S", 0, 



= 0. (51) 



V 



COST 1 ' 



m 



0, 



cosr 

 n 



cos r 



:, 0, 

 cos r cos r'' 



», 0, 



B. I. A. PEOC. VOL. IX. 



= 0. (52) 



0 3 



? * 



cos r cos r 

 cos r cos r 



n' 



0, S", 



m' 



cos r f1 cos r"' 



s,- S\ S', o, 



0. (53) 



3r 



