409 



19. Denoting the equation (28), for shortness, by the notation 

 0 ft m, n, p, q, r,) = 0, and the transverse common tangents by the 

 same notation as the direct common tangents, only with accents, and 

 we have the following seven equations for the other seven pairs of 

 spheres which touch S, S', S", S'", viz.. 



0ft 



m\ 



ri, 



ft 





r,) = 0 



; (29) 



w> 



m, 



n\ 



P> 





r,) = 0 



; (30) 



0 ft, 



m', 



n, 



P> 





r',) = 0 



I (31) 



0ft 



m, 



n, 



P> 





r\) = 0 



; (32) 



0 ft, 



m', 



n, 



f, 



4, 



r,) = 0 



(33) 





m\ 



n>, 



P> 





/,) = 0 



(34) 



0 ft, 



m, 



ri, 







/,) = 0 



; (35) 



20. In precisely the same manner as in Art. 3 we derived the equa- 

 tions of the inscribed and exscribed circles of a plane triangle from the 

 equations of the pairs of circles touching three circles, we can derive the 

 equations of the eight spheres which touch the four faces of a tetrahe- 

 dron from the equations of Arts. 18 and 19. Thus the equation of the 

 inscribed sphere is, the faces being x, y, s, w, derived from equation 

 (28) 



0, cos 2 ^{xy), cos 2 cos 2 ^(xw), x 



cos 2 \{yx\ 0, 



cos 



cos 



2 1 



3 2 |(^), COS 2 ^), o, 



COS 



X, 



2 1 



{wx), 



y> 



(zw), 



COS 2 J(«02) 0, 



y 



COS 



2 I 



0, (36) 



and the equations of the seven others are derived from equations 

 (29)-(3S). 



21. Again, in the same way exactly as we derived the equations of 

 the circles in pairs which touch three circles from the equations of the 

 inscribed and exscribed circles of a plane triangle, we might derive 

 the equations of the spheres in pairs which touch four spheres from the 

 equations of the spheres touching the faces of a tetrahedron \ and, in 

 fact, it was in that way I first derived the theorem. 



22. If we form the tangential equation corresponding to equation 

 (23), we find— 



fxvl + v\m + \fin + \pp + /upq + vpr = 0. (37) 



This is the condition that the pair of spheres given by equation (28) 

 may be touched by the sphere \S + ^S' + vS ff + pS"'= 0. We get si- 



