278 



from a more ordinary form of the equation of the sphere, re- 

 ferred to the edges a ft' y' as oblique coordinates; and, doubt- 

 less, has been already obtained in that or in some other 

 way. An analogous theorem for the ellipsoid may be ob- 

 tained with little difficulty. 



If we suppose in the formula (6), that the point e of the 

 pentagon approaches to the point a, the side ea tends to be- 

 come an infinitely small tangent to the sphere ; and thus we 

 find that V. ABCDA, or that the vector part of the continued 

 product abXbcxcdxda, of the four sides 0/ an uneven (or 

 gauche) quadrilateral abcb, if determined by the rules of 

 multiplication proper to this calculus, is normal to the cir- 

 cumscribed sphere at the point a, where the first and fourth 

 sides are supposed to meet. By the non-commutative cha- 

 racter of quaternion multiplication, we should get a different 

 product, if we took the factors in the order bc x cb x da x ab ; 

 and accordingly the vector or graramic part v . bcdab of this 

 new quaternion product would represent a new line in space, 

 namely, a normal to the same sphere at b : and similarly may 

 the normals be found at the two other corners of the quadri- 

 lateral, by two other arrangements of the four sides as factors. 

 To determine the lengths of the normal lines thus assigned, we 

 may observe that if a', b', c', d' be the four points on the 

 same sphere, which are diametrically opposite to the four 

 given points a, b, c, d, then the four diameters a'a, b'b, c'c, d'd 

 are given by four expressions, of which it may be sufficient to 

 write one, namely : 



V . ABCDA 



A A 3: . (13) 



S . ABCD ^ 



The denominator of this expression denotes (as was re- 

 marked in a former communication) the sextuple volume of 

 the pyramid, or tetrahedron, abcd; it vanishes, therefore, 

 when the four points a, b, c, t> are in one plane : so that we 

 have for any plane quadrilateral the equation, 



s .abcd = 0. (14) 



