194 
ADDITION TO THE MEMOIR ON GEODESIC LINES, 
[511 
diminishes towards zero), the integral K' alters its value only slowly, increasing towards 
a certain constant limit; but, contrariwise, K increases without limit, its value for any 
small value of m being of the form A — B log m, = oo in the limit; wherefore, as m 
diminishes, the value of the am pli tu de of the geodesic, continually increases. If 
this is =1, the geodesic touching at infinity a certain hyperbolic curve of curvature, 
in descending to touch the oval curve, makes round the hyperboloid a half-convolution, 
and then again ascends through another half-convolution to touch at infinity the same 
hyperbolic curve of curvature; viz. it makes in all one entire convolution, or say in 
descending it makes a half-convolution. But if K -=- 2K' = 2, then the curve makes in 
descending a complete convolution; and so, if K -r- 2K' = 2s, then the geodesic makes 
in descending s convolutions; and, as already mentioned, ultimately when m = 0 the 
geodesic makes an infinity of convolutions; that is, it never actually reaches the elliptic 
principal section, but has this line for an asymptote. 
51. To sustain the foregoing statements, I write 6' (=c' + m) = 1600 + m, and I 
consider the integral 
u + 1600 + m 
'00 
du 
K' m = 100000 
(u + 2500 + m) (u + 2000 + m) (u + m) u ’ 
0 
say for a moment this is 
Supposing m to be small, we divide the integral into two parts, say from 0 to a 
[where a, = for example 50 or 100, is large in comparison with m, but small in 
comparison with the numbers (c, &c.), 1600, &c.], and from a to oo. In the second 
part, the expression under the integral sign and the value of the integral varies slowly 
with m, and we may, as an approximation, write m = 0. We have thus 
and the first part hereof is 
viz. the integral is here 
h.l. [u + \m + Vm (ît + m)} = h.l. 
or say this is 
a + i m + Va (a + m) 
= h.l. — approximately ; 
The first term is thus 
which is 
= 4119 log—, 
5 m