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w, or the part of the quaternion (8) which is independent of 
the characteristics ijk, vanishes, so that we have the following 
equation, which is entirely freed from those symbolic factors, 
w= 0, (9) 
we shall then know that the points, of which the rectangular 
coordinates are respectively (4,2) (@2Y2Z2) (@3Y323) (@sY424) 
(5 Ys525), are five homospheric points, or that one common 
spheric surface will contain them all. 
The actual process of this multiplication and reduction 
would be tedious, nor is it offered as the easiest, but only as 
one way of forming the equation in rectangular coordinates, 
which is here denoted by (9). A much easier way would be 
to prepare the equation (5) by a previous development, so as 
to put it under the following form : 
eo’ S.aBy = a’s. Byp + B's. yap + y's. ap; (10) 
which also admits of a simple geometrical interpretation. For, 
by comparing it with the following equation, which is in this 
ealeulus an édendical one, or is satisfied for any four vectors, 
a, B, ys p 
p8-aBy = a8-Byp + BS. yap + y8-aBp, (11) 
we find that the form (10) gives 
p? = aa’ + B' + yy’, (12) 
if a’, B’, y’ denote three diverging edges of a parallelepiped, 
of which the intermediate diagonal (or their symbolic sum) is 
the chord p of a sphere, while a3 y are three other chords of 
the same sphere, in the directions of the three edges, and 
coinitial with them and with p; so that the square upon the 
diagonal p is equal to the sum of the three rectangles under 
the three edges a’ 2’ y and the three chords aBy, with 
which, in direction, those edges respectively coincide. This 
theorem is only mentioned here, as a simple example of the 
interpretation of the formule to which the present method con- 
ducts; since the same result may be obtained very simply 
