285 
Again, it may be remarked, that the condition for a fifth 
point & being contained on the plane which touches, at a, the 
sphere circumscribed about the tetrahedron aBcp, is expressed 
by the equation 
: S.ABCDAE = 0; (28) 
this equation, therefore, ought not to be compatible with the 
equation (6), which expressed that the point £ was on the sphere 
_ itself, except by supposing that the point & coincides with the 
point of contact a; and accordingly the principles and rules of 
this notation give, generally, 
S.ABCDEA + S.ABCDAE = S. ABCD. AEA, (29) 
in which by (14) the first factor s.ancp of the second 
member does not vanish if the sphere be finite, that is, if the 
volume of the tetrahedron do not vanish, while the second 
factor may be thus transformed, 
AEA = — (za), (30) 
_ so that the coexistence of the two equations (6) and (28) of a 
_ sphere and its tangent plane, is thus seen to require that we 
shall have 
am EA =0; (31) 
which i is, relatively to the sought position of £, the equation 
of the point of contact. These examples, though not the 
i m 10st important that might be selected, may suffice to show 
- that there already exists a calculus, which may deserve to be 
further developed, for combining and transforming geometrical 
[ ., M _ expressions of this sort. Several of the elements of such a 
- calculus, especially as regards geometrical addition and sub- 
4 traction, have been contributed by other, and (as the author 
nae believes) by better geometers; what Sir William 
a Hamilton considers to be peculiarly his own contribution to 
____ this department of mathematical and symbolical science con- 
sists in the introduction and development of those conceptions 
