FA. P. Barnard on the Zodiacat Light. 225 
the earth’s shadow. The axis of the light being also visibly par- 
allel to the ecliptic, we need but a single observation of the ze- 
nith distance of the summit to enable us, with a knowledge of 
our latitude and the sun’s hour angle and place in the ecliptic, to 
determine the distance in miles. A general.method for this pur- 
pose may be sketched out as follows : 
The intersection of the horizon with = 
the conical shadow of the earth is an el- 
ipse, of which, according to a well ¥ 
known and general property of conic 
sections, the point of contact with the 
earth, or the observer’s place, is a focus, 
Now if, to an observer at O, the centre 
of the light at its base be in the direc- 
ee OP, and the summit in the direction 
, then the plane OPQ intersects the 
convex surface of the shadow in a line sensibly straight, PQ. 
if we regard the shadow asa cylinder—as for the purposes of 
this inquiry we may do without appreciable error, PQ will be 
truly straight and parallel to the axis of the shadow. 
‘The latitude and the hour angle enable us to find the sun’s de- 
Pression below the horizon, which is the measure of the inclina- 
ion to the horizon of the axis of the shadow, which we may 
represent by the letter I. If we put D for the earth’s diameter, 
then the major axis, HR, of the ellipse, will be equal to D x cosec. I, 
and the minor axis will be D itself. 
The sun’s place in the ecliptic, combined with the data above 
Mentioned, will give the difference of azimuth of the point P 
and the sun, or the angle HOP, and the inclination of the plane 
OPQ to the horizon. ‘The observed vertical altitude of Q, com- 
bined with this last, will give the angle POQ. The axes of the 
ellipse and the azimuthal angle HOP enable us to determine the 
length of the focal radius, OP, From the same azimuthal angle 
and the angle I, we deduce the inclination to the horizon also of 
4 plane touching the cylinder in PQ, of which TN is the trace ; 
and also the angle QPN. ..'Then, at P, we have a solid angle con- 
‘ained by the tangent plane just mentioned, the plane of the hori- 
Zon and the plane OPQ, in which one of the containing plane 
angles and two of the inclinations are known, Hence we derive 
the plane angle QPO, giving, us finally, in the triangle OPQ, two 
i) 
augies and a side, from which OQ is ascertained ; and of this the 
inclination to the horizon has.been found by observation. ‘T 
ce of the point @ from the centre of the earth 18 therefore 
determined, and consequently the altitude of the superior stratum 
of the ring, ies 
Stoono Sxeize, Vol. XXI, No, 62.—March, 1856. 
