476] ORBIT OF A PLANET FROM THREE OBSERVATIONS. 475 
is cut by the meridian, and thus lies partly on the left, and partly on the right 
thereof. 
136. Now by what precedes there is on the separator boundary B'B 1V of the 
lateral region a point where T 13 has a minimum value less than 1-148, and con 
sequently, for any given value, say for a value between this minimum and 1-377, there 
are on B'B ir two points where T 1S has the given value. These points cannot lie on 
the loop of the curve belonging to the given value (for this loop is wholly on the 
left hand of the meridian); hence the complete curve for the given value of T 13 will 
include (within the lateral region) besides the loop, a branch uniting the two points 
in question; say a link branch. 
137. It follows that there is between T 13 = 1377 and 1-983, a value (to fix the 
ideas, say = 1'80 ?, it being understood that I do not attempt to determine this value) 
for which the loop and link branch will unite themselves together, the point of 
junction becoming as usual a node; viz., there will be a curve T 13 = T80 ? having in 
the two lateral regions respectively the nodes Y, Y'; or say the curve has in each 
lateral region a self-intersecting loop. For any greater value of T 13 (as for example 
the value T983 belonging to the cuspidal curve) there are two branches inclosing the 
self-intersecting loop; for a less value, as has been seen, instead of the self-intersecting 
loop, there is a loop and link branch; at least this is the case until for the minimum 
value < 1-148 of T 13 on the separator boundary B 1Y B' the link branch disappears. For 
smaller values down to T 13 = '950, which belongs to the nodal isochronic, there is no 
link branch, but only the loop; and as T 13 diminishes below this value, there is still 
a continually diminishing loop, lying wholly on the left hand of the meridian, and 
with its upper branch always touching the separator; and ultimately for T 13 = 0 the 
loop vanishes. 
138. We have attended wholly to the lateral regions; but the consideration of 
the axial and inner regions is very easy: for any value between the values 1'983 and 
"950, there are in the axial region (between the nodal and cuspidal curves) two 
branches each proceeding from the separator to A, where they unite, and, crossing 
each other, pass into the inner region, forming a loop within the loop of the nodal 
isochronic; and, moreover, there is in the inner region a branch, the continuation of 
the lower branches of the lateral loops, uniting the points A', A", and lying between 
the nodal and cuspidal isochronic. And for T 13 less than -950 there are in the axial 
region, between the nodal curve and the separator, two branches, each proceeding from 
the separator to A, where, crossing each other, they enter the inner region passing 
outside the nodal curve (or in the side regions of the inner region) to the points 
A', A", where they respectively join on to the lower branches of the lateral loops. 
Ultimately, for T 13 = 0, the curve coincides with the finite portions AA', A A" of the 
regulator circle. 
139. We have finally to consider the case T 13 greater than 1’983: there is in 
the axial region a branch lying outside the cuspidal curve, and extending from 
separator to separator; in each lateral region two branches (lying outside those of 
the cuspidal curve) each proceeding from A' (or A") to the separator boundary B'B IV 
60—2