ig ieee 
xliii 
we may write the equation (37) under the form 
o= $e @ 4a) 4 (85 = (40) 
This last equation shows that 
when de — 2 = 0, then & +a?= 0; (41) 
that is to say, when 8 is co-axal with, or parallel to «, or, in 
other words, when the vector from the centre coincides (in 
either ey with the axis of revolution of the surface, its 
length is = + a, according as a is > or < 0. 
The equation (37) shows that 
when dc + 26 = 0, then & 4+ a? (1 + &) = 0; (42) 
if therefore <? 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(—-S)=ayv(l—-e)=b, (43) 
b being a new scalar quantity; but ife?>1, °< — 1, 14+°<0, 
then we shall have, by (42), the absurd result of a vector 8 
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 
a (deal 0; (44) 
and nowhere penetrates within that inscribed spheric surface, 
which has for its equation 
*+a= 0, - (45) 
