( 709 ) 
But let us return after the discussion of these particularities, to 
the treatment of the 7, z-projection of the closed curve. 
We have already observed that the point P is not found when 
dp dp 
the surfaces —- == 0 and —-=O do not intersect. Reversely P 
da? dv? 
extends to a curve if the surfaces do not only touch, but intersect. 
We obtain the equation of this closed curve, if we solve the value 
of ; from the equation : 
Vv v 
nn one = 5) = 0 
dp dp 
and substitute it in =O or 
e dx? 
=(. It is simplest to do this in 
d 2 
F v 
v—b\? 
aw a b 
Orato tp 2 
dv? b he 
b 
b b Ma el b 
MRE G Nn ~) . If we write — in the following 
v v 
form: 
or to substitute the value of 
6 1-VA—B+AB_ 1x 
dn Pane) Gor TER 
and 
: Be ee VA—B+AB BtyX 
a ear 1+B Parte 
we find: 
oen 4 + (2B--B’ x 
Mn eee ae es ET a 
b (1+ B) 
When X= A—B+AB=AB 5-2 — i is positive, 7’ is 
real, and there are two values of 7’ for every value of rz. For the 
same values of v for which in the v, v-projection the two values of 
v . . > . . . r, . ° 
7 coincide, the two values of 7’ coincide in the 7’, z-projection. 
The values of 7’ assume a simple form for these limiting values 
r ryy B , . 
of «x, because then VX =O, i MRT = 2 - 5 or TB of course this 
value must also hold for the case that these limiting values of w 
coincide, which we treated above. We can even simplify this form 
