446 
ON THE DETERMINATION OF THE 
[476 
93. For the calculation of the table we have 
log x x = 10 + log sec c, 
log y x = 10'76144 + log tan c, 
log x 2 = 10-65052 + log sec (c — 26° 34'), 
log (y 2 — '92376) = 10 06247 + log tan (c — 26° 34'), 
log x 3 = 1065052 + log sec (c 4- 26° 34'), 
log (y 3 + -92376) = 16-06247 + log sec (c + 26° 34'), 
the values of r X} r 2 , r 3 , are then calculated from 
= r x cos ф и 2/i = r i sin 0i > 
or say 
— = tan ф х , r x — x x sec ф 1г &c. 
х г 
and those of the chords 7 Ш y 23 , <y 31) from 
X\ — x 2 = Yjo cos ^ 12 , 2/1 — 2/-’ ~ 7i2 sin #i2, 
or say 
tan в 12 = ———, 7 12 = {x x — x 2 ) sec 6 12 . 
2/i — 2/2 
We have then to find the equation of the orbit r = Ax + By+ G; this might be done 
by substituting in the determinant expression the numerical values of x x , y x , r lt x 2 , y 2 , r 2 , 
х з> Уз> г з> an d so calculating the result, but I have preferred to employ the formula of 
No. 90, using only the calculated values of r l5 r 2 , r 3 \ viz. we have 
v x — Ry, 
v 2 (X. + 2) = R 2 , 
r 3 (X - 2) = R 3 , 
which gives the values of Ry, R 2) R 3 . And then we have e, rs, a, from the equations 
. jf ±G 
A = e cos cr, i) = e sin vr, a= , 
1 — e- 
e and a being each regarded as positive. The times in the elliptic, and parabolic 
orbits are then calculated from Lambert’s equation, as explained in regard to Planogram 
No. 1, but for the hyperbolic orbits, the other formulse were made use of.