285 



Again, it may be remarked, that tiie condition for a fifth 

 point E being contained on the plane which touches, at a, the 

 sphere circumscribed about the tetrahedron abcd, is expressed 

 by the equation 



S . ABCDAE = ; (28) 



this equation, therefore, ought not to be compatible with the 

 equation (6), which expressed that the point E was on the sphere 

 itself, except by supposing that the point e 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 . abcd 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 = — (ea)S (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 



ea = 0; (31) 



which is, relatively to the sought position of e, the equation 

 of the point of contact. These examples, though not the 

 most 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 

 expressions of this sort. Several of the elements of such a 

 calculus, especially as regards geometrical addition and sub- 

 traction, have been contributed by other, and (as the author 

 willingly believes) by better geometers ; what Sir William 

 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 



