472 
ON THE DETERMINATION OF THE 
[476 
128. The times T 12 , T 23 , T 31 are calculated, Planogram 1, part 1, for the meridian 
long. 90°, and ditto part 2 for the meridian long. 270°; and in Planogram 2 for the 
meridian long. 180°. As regards these last values, it is easy to see that, in order to 
pass to the meridian long. 0°, the numbers 2, 3 must be interchanged; that is, 
long. 0°, the T 12 , T 13 , T 23 are respectively equal to the values, long. 180°, T 13 , T 12 , T 23 . 
Moreover, the numbers 1, 2, 3 may be changed into 2, 3, 1, or into 3, 1, 2, provided 
the longitude is increased by 120° and 240° in the two cases respectively; that is, 
T 31 long, a = T 31 long, a 
= T 12 long, (a + 120°) 
= T 23 long, (a + 240°). 
129. By means of the foregoing two relations, T 13 for the several longitudes 
0°, 30°, 60°, ... 330°, is given as equal to the T 12 , T 23 , or T 31 , for long. 90°, 270°, or 
180°, that is, to the T 12 , T 23 , or T 31) of Planogram No. 1, part 1 or 2, or of Planogram 
No. 2. For example, T 31 long. 240° = T 12 long. 0° = T 13 long. 180°, that is, it is equal to 
the T 31 of Planogram No. 2. We thus find 
Long. T 13 is = 
0° . . . T 12 of Plan. No. 2 
30° T 23 of Plan. No. 1, pt. 2 
60° . . . T 12 
90° T 12 of Plan. No. 1, pt. 1 
120° . . . T n 
150° T 31 
180° . . . T 31 
210° T 23 
240° . . . T u 
270° T 12 
300° . . . T n 
330' T 31 
and observing that for Planogram No. 1, part 1 or 2, we have T 12 = T 31 , it hence 
appears as above, that the meridian 30°—210° is an axis of symmetry of the 
spherogram. In what precedes it has been assumed that the colatitudes only extend 
from 0° to 90°, but in the spherogram they extend for the meridians 30°, 150°, 270°, 
to the colatitude 106° 6', the values for the colatitudes above 90° are those for the 
omitted portions 90° to 73° 54' of the opposite meridian. 
N.B. A meridian extends from the pole in one direction only, unless the contrary 
is expressed or implied, as in speaking of a meridian 0°—180°.