476] ORBIT OF A PLANET FROM THREE OBSERVATIONS. 471 
125. We may without difficulty attach to the several portions of the regulator, 
the separators and the parabolic curve, to each portion its proper symbol L, P, H 
and 123, 1.23, &c. as the case may be. 
First, as to the regulator, it is obvious that this is separated by the points A 
into the three portions L 213, L 321, L132, respectively. And inside the regulator, 
adjacent to these, we have portions of the parabolic curve P 213, P 321, P 132, 
respectively. 
Again, for one of the separators, say B 1V B'AB"B V (see here and in all that follows 
the notation-diagram, No. 115); since the point 2 is here at infinity this must be at 
every portion thereof either H132 or else H 312. The point B iy is H i32 and the 
point B' is H 312; consequently, as the orbit-pole passes along the separator from 
B iy to B', the symbol is at first H132 and at last R 312 ; the transition takes place 
at the point of contact of the parabolic curve which is indifferently P132 or P 213. 
(In further explanation of the transition, consider the orbit-pole as passing from 
B iy to B, not on the separator, but indefinitely near it; it can only do so by 
twice crossing the parabolic curve near the point of contact; the orbit is first H132, 
or say H132, then P132, then an ellipse, which when the orbit-pole again arrives 
at the parabolic curve changes into P 312; and it finally becomes H 312 or H 312.) 
126. Again, since, on the two separators through B ly , in the portions adjacent to 
P IV , the symbols are H132 and H i32, it is clear that in the adjacent portion of 
the parabolic curve (terminated each way by a point of contact with these separators 
respectively) the symbol must be P132; at the point of contact with the first- 
mentioned separator B iy B'AB"B y , this becomes P132, =P213; and beyond the point 
of contact it becomes P 213, continuing so until it arrives at the next point of contact 
with the separator B'A'B": there is always in the symbol for the parabolic curve this 
change of form as we pass through a point of contact with a separator; and there 
is the same change, when travelling along the loop (that is without going inside the 
regulator) we pass through a point A. The foregoing considerations fully explain how 
the proper symbol is to be attached to each portion of the regulator, the separators, 
and the parabolic curve: to avoid confusion, I have abstained from attaching them in 
the Plate. 
127. Imagining the symbols attached as above, it at once appears that, for the 
two portions A'A and A A" of the regulator curve, we have T 13 = 0; while, for the 
arc A"A' of the parabolic curve we have T 13 = co. Moreover, T l3 can only be infinite 
on one of the separators through B'" and on the parabolic curve; and the symbols 
show that the curve T 13 is made up, in a peculiar discontinuous manner, of portions 
of these two separators and of the parabolic curve, as shown by the strongly marked 
line of the figure ; we have thus the boundary of certain lightly shaded regions within 
which (as well as within the shaded regions) T 13 is non-existent; excluding these, the 
remaining regions (instead of a trilateral symmetry) have a symmetry about the axis 
BB'"; there are still four regions which may be distinguished as the inner region, the 
axial outer region, and the lateral outer regions; or, more shortly, as the inner, axial, 
and lateral regions.