277 



IV, 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 (xiPiZt) (x^y^z^) {^ay-i^z) {^iyi^A) 

 {x^y^z^), are Jive homosphceric 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 : 



p^S. aj37 — a^S . j3yp + jS^S.yap + j^ S . aj5p ; (10) 

 which also admits of a simple geometrical interpretation. For, 

 by comparing it with the following equation, which is in this 

 calculus an identical one, or is satisfied for any four vectors, 

 a? /3, 7, p : 



/oS . a/By = as . (5yp + /3s .yap + yS .aj3|0, (11) 



we find that the form (10) gives 



p^ = aa'-\- (515' +yy', (12) 



if a', /3', 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 ajSy 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'jS'y' and the three chords ajSy, with 

 which, in direction, those edges respectively coincide. This 

 theorem is only mentioned here, as a simple example of the 

 interpretation of the formulae to which the present method con- 

 ducts ; since the same result may be obtained very simply 



