ON GEODESIC LINES, 
166 
[508 
all touch the curve of curvature determined by the parameter 0, that is the curve 
which is the intersection of the surface by the confocal surface 
X? if Z 2 
a+ 0 b + 0 c + 0 
15. In the particular case 0 = oo, the equation becomes 
Pdp 2 — Qdrf — 0, 
that is 
pdp? 
qdcf 
= 0, 
(a+p)(b+p)(c+p) (a + q)(b +q) (c +q) 
which is the differential equation of the circular curves on the surface. 
16. The signification of the case 0 = 0 is not at first sight so obvious. Supposing 
that 0 is first indefinitely small, and writing the equation in the form 
we have the series of geodesics touching the (imaginary) curve of curvature, the 
QQ 2 qj2 gl 
intersection of the surface by the imaginary cone — + ~ + — = 0. These are, in fact, 
Of 0~“ Of 
the right lines on the surface: I apprehend that the intersection in question is not 
a proper envelope, but is the locus of nodes of the geodesics, viz. each geodesic is 
to be considered as a pair of lines belonging to the two sets: I do not, however, 
quite understand this. 
17. I say that the geodesics in question are the right lines on the surface; 
viz. writing in the differential equation 0 = 0, it is to be shown that the differential 
equation of the right lines is 
dp dq 
= 0, 
— r —I— ■* 
f{a + p) (b + p) (c+p) V (a +q)(b + q) (c + q) 
or what is the same thing, that the integral of this equation represents the right 
lines on the surface. 
Writing the equation of the surface in the form 
a b 
t_ i _i 2 
b e' 
x iy _ 1 
fa fb cr 
we have at once