511] 
IJST PARTICULAR THOSE OF A QUADRIC SURFACE. 
193 
which was used for the values w = 9, 8, 7, ... 1, 0, to complete the calculation up to 
q = — 900. 
47. We have in like manner, p = lQoQ + u, 
n (p)= 1 ooooodu ^ + 255Q) ^ +2050 ^ ^ + 50) u , 
which for small values of u is 
- 100000 tj 2550 , 50 (/ ^ ■ = 2 V»); 
viz. this is 
502-5 (log = 2-7011399) Vw, 
used for w=l, 2, ... 10, that is to p = 1660. The calculation was afterwards continued 
by quadratures, by giving to u a succession of values, at intervals at first of 10, and 
afterwards of 20, 50, 100, 200, and 500, up to p — 10,000, giving for the integral the 
value 10411; and thence, as appearing above, the value for p—oo was found to be 
= 12490. 
48. After the calculation of the values of II (p) and T (q), it was easy by inter 
polation to revert these tables, so as to obtain a table which, for II or ^ as argument, 
gives the values of p and q. The arguments are taken at intervals of 500; up to 
10000 as regards p, since the original table was only calculated thus far; and up to 
34726 as regards q. I had thus calculated the annexed Table III., when it occurred 
to me that there was a convenience in taking the arguments to be submultiples of 
the complete integral 34726; say we divide this into 90 parts, or, as it were, graduate 
the quadrant of the hyperboloid by means of hyperbolic curves of curvature adapted 
for the geodesics in question. Taking every fifth part, or in fact dividing the quadrant 
into 18 parts, we have the Table IY. 
49. It will be remembered that the foregoing results apply only to the geodesics 
which touch the oval curve of curvature p — + 1650; for the geodesics touching any 
other oval curve of curvature, the values of the integrals, and the mutual distances 
of the curves of curvature used for tracing the geodesics, would be completely altered. 
But it is possible to derive some general conclusions as to the geodesics that touch 
a given oval curve of curvature. 
Observe that the integral A 7 (= 34726 in the case considered) measures the 
quadrant of the hyperboloid ; viz. 'T' (q) = 0, T r (q) = K' determine two hyperbolic curves 
of curvature (principal sections), the mutual distance whereof is a quadrant. Each 
geodesic touches the given oval curve of curvature, and it touches at infinity the two 
hyperbolic curves "'B (q) = Q ± K (K — 12490 in the case considered); viz. the distance 
of these in regard to the circuit of four quadrants, or say the amplitude of the 
geodesic, is measured by the ratio • 
50. Now it is easy to see that as the oval curve of curvature approaches the 
principal elliptic section, that is, as 6' approaches c' (or writing 6' = c' + m, as m 
c. viii. 25