126 PHILOSOPHICAL SOCIETY OF WASHINGTON. 
The equation of this ruled surface is obtained by eliminating 2, y, z 
from (1), (2), and (5). It is important to remark that the dior- 
thode does not consist of parts which are diorthodes with respect to 
their termini, otherwise the normals would at the same time pass 
through two chords from the same point and the curve would be a 
plane curve. Dr. Bremiker had erroneously supposed that the 
diorthode was touched by the normal planes. This is only the case 
at the termini. He has been criticized by Dr. Bruns of Pulkowa 
and by Helmert, but neither critic has shown the existence of a curve 
possessing this property, namely, the prodrthode, in which the nor- 
mal plane at any of its points passes through the consecutive point 
and the forward terminus, but not in general through the starting 
point. If then in (5) we replace (,, y,,2,) by («+ dz, y+ dy, 
z+ dz) we have: 
0=[y.—y) de— (e,— 2) ay (FZ) 
+[@—2)dy— (de ($2) 
+ [(a,— x) dz — (z,— z) dx] (3 ai 
Gi 
aaa s>2) 7) 1@ 
+[@-9 (SE) =< a -»(¢ a) |e 
du U 
+[@- (F) -% —» (Z) Je 6) 
By means of the equation of the surface (1) and its differential 
equation 
d d x 
o= (34) de + (FH) ay + (FE yd) 
any one of the variables with its differential can be eliminated. 
The resulting differential equation being integrated so as to contain 
the starting point (2, y,, 2), will be the equation of a projection of 
the prodrthode on a coérdinate plane. 
The proérthode being differently related to its ends, will be dif 
ferent furward and backward, while the diorthode is the same for- 
ward and backward. 
