xliii 

 we may write the equation (37) under the form 



Orr(l+r)(S^ + fr) + (^^y. (40) 



This last equation shows that 



when Se - fS z= 0, then 8'^ + a^ — o ; (41) 



that is to say, when S is co-axal with, or parallel to t, or, in 

 other words, when the vector from the centre coincides (in 

 either direction) with the axis of revolution of the surface, its 

 length is = ± a, according as a is > or < 0, 

 The equation (37) shows that 



when 8£ + eS = 0, then l^ + a^l + e^) = ; (42) 



if therefore e^ be > — 1, that is, if e'^ < 1, the length of every 

 vector drawn from the centre perpendicularly to the axis of 

 revolution will be 



V(- g^) = a y/{\ -e") = b, (43) 



ftbeing a new scalar quantity; but if e^>l, £'^< - 1, l+£^<0, 

 then we shall have, by (42), the absurd result of a vector S 

 appearing to have a positive square : whereas it is a first 

 principle of the present method of calculation, that the square 

 of every vector is to be regarded as a negative number : which 

 symbolical contradiction indicates the geometrical impossi- 

 bility of drawing from the centre to any point of the locus^ 

 a straight line which shall be perpendicular to the axis of re- 

 volution, in the case where e^> 1. The locus has, in this 

 case, two infinite branches enclosed within the two branches 

 of the asymptotic cone which has for its equation 



and nowhere penetrates within that inscribed spheric surface, 

 which has for its equation 



g2 + a^ = 0, (45) 



