192 
ADDITION TO THE MEMOIR ON GEODESIC LINES, 
[511 
log* oq 
and hence, substituting in the formula, for h.l.x its value , 6 , the superior limit is 
log e 
vs 
/, + J log + W* cot-. M, 
(y + Sloge ®VPi — VS y + S VtJ 
and the inferior limit is 
/,+j V V ,JL log + 2 cot- V4 • 
(a +/Stage °Vpi —V/3 a + V a i' 
45, The numerical values are y> x = 10,000; a, b, c, 6'= 900, 400, 1600, 1650; and 
thence determining by trial values of m and n, 
a = 650, 7 = 650 — 4 = 646, 
/3= 1625, 5 = 1625 + 160 = 1785, 
I obtained for the logarithmic and circular terms of the two limits respectively 
Superior Inferior 
Logarithmic '015668 ‘015144 
Circular ‘005202 ‘005593 
•020870 ‘020737 
The value of was 10411 = 100,000 = T04110, and the two limits thus are T24980 
and T24850; or restoring the factor 100,000, they are 12498 and 12485; the mean of 
these, say 12490, was taken for the value II (p), p = oc ; that is K = 12490. 
46. As regards the calculation of the integrals II (p) and T (q), introducing the 
numerical values, and multiplying by the before-mentioned factor 100,000, we have 
(q = — 400 — v), 
'T (q) = 100,000 civ sj(500 _ v) v <+ + 2000) (v + 2050) 5 
which for any small value of v is 
viz. this is 
= 100,000 
y 
400 
500.2000.2050 
dv 
yV 
= 883-45 (log = 2-9461830) 05 
which was used for the values v = 1, 2, ...10, that is, to q = — 410; after which the 
calculation was continued by quadratures giving to v the values 10, 20, 30,... up to 
v = 490, or q— — 890. For the remainder of the integral, writing 500 —v = w (that is, 
q = — 900 + w), we have 
*<*>-* ( - 890) “ 10 °' 000 h V W (500- W )(2 9 500Zm50^) ’ 
= 10 °- 000 \/5U0 = 5iT256O 1/S ‘ -WW)} 
= 1062-7 (log = 3-0264261) (V10 - Vw),