511] 
IN PARTICULAR THOSE OF A QUADRIC SURFACE. 
195 
or we have 
An f 00 
K' m — 4119 log — + / U 0 du. 
VI J a 
s 
We ought to have the same value of the integral, whatever, within proper limits, 
the assumed value of a may be. Taking, for instance, a = 50 and a = 100, we ought to 
have 
/, 
K' m = 4119 log—+ 
° on 
U 0 du 
100 
that is, 
4119 log 2 = 
In verification, I calculated the second side by quadratures; viz. for the values 
50, 60, 70, 80, 90, 100, the values of U 0 are 35-532, 29-570, 25-311, 22'373, 19'632, 
17"645; whence, adding the half sum of the extreme terms to the sum of the mean 
terms, and multiplying by 10, the value of the integral is = 1234'74. The value of 
the left-hand side is = 1239*94, which is a sufficient agreement. 
52. Returning to the formula for K' m , this may be written 
I did not calculate the value of the integral in this formula, but determined the 
term in ( ) in such wise that the formula should be correct for the foregoing 
value m = 50 ; viz. the term thus is 
= 12490 + 4119 log 50 = 12490+ 6998, =19488, or say 19500; 
we thus have 
K m = 19500 — 4119 log m ; 
and we may roughly assume that, for any small value of on, K' m has the same value 
as for on = 50; viz. we may write 
K' m = 34726, or say =35000. 
We thus see how to give to on such a value that the quantity which is 
the number of convolutions of the geodesic, may have any given value; and, in 
particular, we see how exceedingly small on must be for any moderately large number of 
convolutions; for instance, on = ^qq 000000 ° r ^ m “ “ 8, K = 19500 + 32952, = say 52500. 
or the number is = , about five-sevenths of a convolution. 
Correction. Instead of speaking, as above, of a geodesic as touching at infinity 
a hyperbolic curve of curvature, the accurate expression is that the geodesic at infinity 
is parallel to a certain hyperbolic curve of curvature. The geodesic has, in fact, for 
asymptote the right line on the surface parallel at infinity to such curve of curvature. 
Added Dec. 1873. 
25—2