281 



It may seem strange that the line and circle should here be 

 represented each by only one equation ; but these equations 

 are of tJecfor_^r»?«, and decompose themselves each into three 

 equations, equivalent, however, only to two distinct ones, 

 when we pass to rectangular coordinates, for the sake of com- 

 parison with known results. 



In the same notation of capitals, whatever five distinct 

 points may be denoted by a, b, c, d, e, we have the general 

 transformation, 



ABCDEA =: ABCA X ACDA X ADEA -^ ACADA, (18) 



in which the divisor acada, or aca X ada, is the product of 

 two positive scalars ; if then we had otherwise established 

 the interpretation lately assigned to the symbol abca, as de- 

 noting a line which touches at a the circle abc, we might 

 have in that way deduced the equation (6) of a sphere, as the 

 condition of the coplanarity of the three tangents at a, to the 

 three circles, abc, acd, ade. And we see that when this 

 condition is satisfied, so that the points a, b, c, d, e are homo- 

 sphaeric, and that, therefore, the symbol abcdea represents a 

 vector, we can construct the direction of this vector by draw- 

 ing in the plane which touches the sphere at a, a line Ai A2 

 parallel to the line acda whicji touches the circle acd at a, 

 and cutting, in the points Aj and Ag, the two lines abca and 

 ADEA, which are drawn at a to touch the circles abc, ade ; 

 for then the vector abcdea, which is thus seen to be a tan- 

 gent to the sphere, will touch, at the same point a, the circle 

 a A1A2, described on the tangent plane. In the more general 

 case, when the condition (6) is not satisfied, and when, there- 

 fore, the rectilinear pentagon abcdb, which we shall suppose 

 to be uneven, cannot be inscribed in a sphere, the scalar symbol 

 s'. abcdea which has been seen to vanish when the pentagon 

 can be so inscribed, represents the continued product of the 

 lengths of the Jive sides ab, bc, cd, de, ea, multiplied by the 

 sextuple volume of that triangular pyramid ivhich is con- 



