174 
ON GEODESIC LINES, 
[508 
or, what is the same thing, 
, ... ,'dcr dr 
v (i + ^ (VS + VT. 
i (i> + 6) 
da 
dT 
7r 
= 0; 
that is, 
(p-q)8 ^ ) + 2 [ 2 P2 + ^ (i> + 2)] = 0; 
or substituting herein for p — q, pq, p + q the values V2T, B, A, each divided by 
(a + t) 2 , this is 
0 (Tda* + Xdr 2 ) + 2(2B+ 6A) dadT = 0, 
or say 
6 (Tda 2 + 2Adadr + Xd+ 2 ) + 4Bdadr = 0 ; 
viz. writing herein 6 = 0, the equation is da dr = 0, giving the right lines on the 
surface; and writing 6 — co, it is Tda 2 + 2Adadr+ Xdr 2 — 0, giving the circular lines. 
29. The equation ds 2 = Tda 2 + 2Adadr+ Xdr 2 -i-(t + a) 4 shows that the right lines 
a, a + da, r, t + dr form on the surface an indefinitely small parallelogram, the sides 
whereof are yT da + (T + a) 3 and y/X da 4- (t + a) 2 , viz. the ratio of the coefficients of 
da, dr is of the form function a function t ; and it thus appears that it is possible 
to draw on the surface the two sets of right lines, the lines of each set being at 
such intervals that the surface is divided into parallelograms, the sides of which have 
to each other any given ratio (the angles being variable); viz. if this ratio be as 
m : 1, then, to determine the relation between a, r, we must have yT da — ± m dr, 
or what is the same thing, = ±m ~ 
\/2 v 1 
will be rhombs; and we must then have 
In particular, if m = 1, the parallelograms 
da dr 
yl =± yT ; 
viz. this being in terms of a, r, the differential equation of the curves of curvature, 
it appears that the two sets of lines may be taken so as to divide the surface into 
indefinitely small rhombs, such that, drawing the diagonals of these, we have the two 
sets of curves of curvature. 
The Ellipsoid and the Skew Hyperboloid. 
30. I have thus far considered a quadric surface in general, the various theorems 
being applicable as well to the ellipsoid and the hyperboloid of two sheets as to the 
skew hyperboloid, the right lines being of course imaginary for the first-mentioned 
surfaces; but I will now consider the ellipsoid and the skew hyperboloid separately. 
31. First the ellipsoid. We have here a, b, c all positive, and I assume as 
rjQ2 qpi 
usual a >b> c. The principal sections are all ellipses, viz. — + — 1 is the major-