476] 
ORBIT OF A PLANET FROM THREE OBSERVATIONS. 
473 
130. I attend, in the first instance, to the axis of symmetry or meridian 30°—210°, 
Proceeding along the meridian long. 30° or towards the point A, the value of T 13 
decreases from 1 at the centre to a minimum ='950 at colatitude 11° (call this the 
point X), and it then increases to T983 at A, and thence to 58'62 at 90° and oo 
at the parabolic boundary of the axial region. In the opposite direction it increases 
from 1 at the centre to oo at the parabolic boundary of the inner region. The 
minimum value '950 on the axis of symmetry indicates a node on the isochronic 
curve; that is, the point X is a node on the isochronic T 13 = '950. This will consist 
of two branches, proceeding from A', A", respectively, cutting the axis and each other 
at X, then again cutting at A, and thence passing on into the axial region, and 
respectively terminating on the separator boundary B'AB" thereof. 
131. This curve, which I call the nodal isochronic, divides the inner region into 
a loop, antiloop, and two side regions. On each of the meridians 0°, 60°, the value 
of T 13 diminishes from 1 at the centre to a minimum which is less, and then 
increases to a maximum which is greater, than '950; the value then diminishes to 0 
on the regulator: on emergence of the meridian from the shaded into the axial 
region, the value is = '909, and it thence increases to go at the parabolic boundary 
of the axial region; these data further determine the form of the nodal isochronic, 
viz., each of the two half meridians cuts the loop twice, and again cuts the curve in 
the axial region. 
The nodal isochronic, at each of the points A', A", continues its course into the 
lateral region, returning to the same point A or A', so as to form in each of the 
lateral regions a loop. Considering the loop as formed of two branches, each proceeding 
from A' or A", the one which is the continuation of the course within the inner 
region I call the lower branch; the other, the upper branch; and I say that the 
upper branch touches the separator at A' or A". The two branches and the entire 
loop lie on the left-hand side (or side away from A) of the meridians through 
A' or A". As to the contact of the upper branch of this and other isochronics at 
A' or A" with the separator, see post No. 142. 
132. It is convenient at this point to consider the form of the isochronic curves 
within the axial region. The parabolic boundary thereof is an isochronic T 13 = oo, and 
it thence appears that for any large value of T 13 the isochronic curve (portion of the 
curve) is a curve not meeting the parabolic boundary, and terminated each way in 
the separator boundary B'AB". As the value of T Vi diminishes, the curve (which is 
of course always symmetrical in regard to the axis) bends inwards towards the point 
A and for T 13 = 1'983 (value on the axis at A) the curve acquires a cusp at A. 
I call this the cuspidal isochronic; I remark that it intersects in the axial region 
each of the meridians 0° and 60°. 
As T l3 further diminishes to any value between T983 and '950, the curve, 
commencing in the separator boundary, passes through A into the inner region, and, 
forming a loop within the loop of the nodal isochronic, emerges through A into the 
axial region, terminating again in the separator boundary. 
C. VII. 
60