a 
281 
It may seem strange that the line and circle should here be 
represented each by only one equation; but these equations 
are of vector forms, 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 4, B, ¢, D, E, we have the general 
transformation, 
ABCDEA = ABCA X ACDA X ADEA — ACADA, (18) 
in which the divisor acaba, or AcA X apa, is the product of 
two positive sealars; 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 anc, 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- 
spheric, and that, therefore, the symbol aBcpEa 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 A, Ao 
parallel to the line acpa which touches the circle acp at a, 
and cutting, in the points a, and a,, the two lines aca and 
ADEA, which are drawn at A to touch the circles aBc, ADE; 
for then the vector aBcpEA, which is thus seen to be a tan- 
gent to the sphere, will touch, at the same point 4, the circle 
A A; Ao, described on the tangent plane. In the more general 
case, when the condition (6) is not satisfied, and when, there- 
7 fore, the rectilinear pentagon ascprE, which we shall suppose 
to be uneven, cannot be inscribed in a sphere, the scalar symbol 
$.ABCDEA which has been seen to vanish when the pentagon 
can be so inscribed, represents the continued product of the 
lengths of the five sides aB, BC, CD, DE, EA, multiplied by the 
sevtuple volume of that triangular pyramid which is con- 
