xlii 



y' being a new disposable scalar, ami S a new variable vector ; 

 and, after having cleared the equation (15) of the radical r or 

 \/ { — a^), by writing it as follows, 



«'+(?- ^7=0. (33) 



we get 



= {g'e +8) ^ + (p + ^V - ?^^)' (34) 



- 82 ^ (!i±i^y + ^"(Se + 6§) + {p+ g'ey - g"e- ; 



if we make for abridgment 



g" -g'~{p + g'e% (35) 



If 7' or g'e is to be the constant vector of the centre of the 

 locus, it is necessary that to every variable vector, 8, which 

 satisfies the equation (34), should correspond another vector 

 — S, equal in length but opposite in direction, and satisfying 

 the same equation ; therefore the terms g"(^t + tS) must dis- 

 appear, and we must have 



g"=0,g'= -j-^^^' W 



the constant a being thus suggested by the search after a cen- 

 tre, as well as by the search after a second focus. Making 

 then g' zz.ain (34), we find the following equation of the 

 surface, when referred to its centre, 



= S^ + (Mj?y4.ap; (37) 



in which 



aj9 = a\\ - e") = a\\ + e''). (38) 



And because in general, -for any two vectors S, t, the fol- 

 lowing relation holds good, 



