474 
ON THE DETERMINATION OF THE 
[476 
133. On the meridians 90°, 330°, through the points B', B", respectively, the value 
of T 13 diminishes from 1 at the centre to 0 at the regulator, where these meridians 
are considered as terminating. 
On the meridians 120°, 300° (meridian at right angles to the axis of symmetry), 
the value of T 13 diminishes from 1 at the centre to a minimum less than '878, and 
then increasing to a maximum of over '895 diminishes to 0 at the regulator. On 
emergence of the meridian from the shaded and half-shaded region on the parabolic 
boundary of the lateral region the value is = go , and it thence diminishes to 1T48 
on the separator boundary B 1V B' or B V B". 
On the meridians 150°, 270°, which pass through A', A", respectively, the value 
of T 13 increases from 1 at the centre to 1377 at the regulator, and thence through 
2255 at 90° to go at B iy or B v . 
And finally, on the meridians 180°, 240°, the value of T 13 increases from 1 at 
the centre to oo at the parabolic inner boundary, and then on emergence from the 
half-shaded and shaded region at the separator boundary B'"A' or B"'A", the value 
is = oo, and it thence diminishes to a minimum under 6'343, and again increases to 
oo at the separator boundary B'"B iy or B"'B V . 
134. By what precedes, it appears that on the separator boundary B IY B' or B y B" 
of either of the lateral regions, the values of T 13 is at each extremity = co, and at an 
intermediate point = 1*148; there is consequently a minimum value less than T148, 
and therefore two points at each of which the value is = T983. 
Now resuming the consideration of the cuspidal isochronic (T 13 = 1'983) as regards 
the remaining portions thereof, viz., those in the lateral and inner regions; and con 
sidering first the lateral region B"'B iy B', there will be from each of the points just 
referred to on the boundary B iy B' a branch; one (which I call the lower branch) from 
the point nearer B', passes, on the right-hand side of the meridian through A', to A'; 
the other (which I call the upper branch) proceeding from the point nearer B Iy , cuts 
the same meridian, and then on the left-hand side thereof arrives at A', touching 
there the separator: at A" in the other lateral region there are in like manner an 
upper and a lower branch (situate symmetrically, in regard to the axis, with the upper 
and lower branches at A'); and continuous with the two lower branches there is a 
branch from A' to A", through the antiloop of the inner region. 
135. Imagine the given value of T i3 as continuously increasing from the value 
'950, which belongs to the nodal isochronic; and attend in the first instance to the 
form within the lateral regions. There will be a loop of continually increasing 
magnitude (viz., the loop for a larger value of T 13 will always wholly include that for 
a smaller value); each loop formed by an upper branch, which at A' touches the 
separator, and a lower branch the direction of which from A' is variable. So long 
as T 13 is less than T377 (value at A' along the meridian) the lower branch, and 
consequently the whole loop, will lie on the left hand of the meridian; but when T 13 
is =1377, the lower branch touches the meridian, and for any greater value of T 13 
lies on the right of the meridian; and in either of the last-mentioned cases the loop