12 
MESSES. DE LA EUE, STEWART, AND LOEWY’S RESEARCHES 
L=NM, 
D (negative) =MC, 
whence, since MNC=I, NC=180°+ O-N, NMC=90°, 
tan L= cos I tan (O — N) (5) 
sin D = sin I sin ( © — N) (6) 
These four auxiliary angles, viz. G, H, L, D, have been calculated for every degree of 
the arguments, viz. © for G, and (©— N) for H, L, and D ; and in these Tables, as 
mentioned in § 4, the elements I and N employed by us are those arrived at by 
Mr. Carrington. 
12. To pass now to the last step in the reduction, let P (fig. 9) be the sun’s north 
pole, N' the celestial pole, and N' C the meridian through the apparent centre of the 
sun ; further, let N D M be the solar equator, N the position of the ascending node, and 
S that of a spot. 
Then NBD=Z is the heliographical longitude of the spot Fig. 9. 
reckoned along the solar equator from N. Also NBDM=L = helio- 
graphic longitude of the earth. 
D S=x is the spot’s heliographical latitude. 
Now, in the triangle P S C the angle P C S is obviously the algebraical 
sum of the angle S C N', or the angle of position of the spot S (=P), 
reckoned from north towards east, and the angle N' C P=G+H. 
Let angle S CP=S C N'+N' CP=P + G-f-H=%, further we have 
SC=f= angular distance from centre, and PC=90°— D. 
Figure 10, which has been suggested to us by Sir John Herschel, places these facts 
in a more general and probably also clearer manner before the reader. In this figure 
the sun’s equator is taken for a plane of projection, and the spherical triangles are re- 
ferred to the centre of the sun. The letters are identical with those in fig. 9, while C 
in fig. 10, looked at from without by a spectator placed in the line of the earth’s situa- 
tion, corresponds to C in fig. 9, the centre of the visible disk. 
The triangle P S C will give us, by well-known Eig. 10. 
fundamental formulae of spherical trigonometry, 
sin x= cos g sin D -fi sin g cos D cos 
sin (L — 1)= sin % sin g sec X, 
from which the heliographical coordinates are at 
once obtained. 
The following page exhibits the final reduction 
of all spots observed on August 25, 1862, the mea- 
surements of which, with their primary reductions, 
have been given on pages 10 & 11 above. On the 
pages 45 et seq. the results of the positional elements 
for the spots observed during 1862 and 1863 are 
given under the following heads : — . 
