A400 F. A. P. Barnard on the Zodiacal Light. 
Put ©’s depression =d, ZQ=e, SQ=9’, ZQS=Q. Then, in 
the triangle ZQS 
sin =cos ¢ cos ¢+sin ¢ sin oe cos Q. 
But, by the hypothesis, e+9’=90°. Consequently cos o’=sin 
nc 
9; and sin &=cos ¢. hence, 
Sin 6=sin ¢ cos 9 (1+cos Q)=¢ sin 2¢ (1+cos Q), 
2 sin 
aes 
Or, cos aca 1. 
When 2 sin 9=sin 2 9, cos @=0, and the ring touches the 
horizon. When 2 sin 6> sin 2, cos Q is positive, and the ob- 
scured part of the ring rises above the horizon. When 2sind< 
sin 29, cos Q is negative, and the light is visible to the observer 
at O 
It is evident that, 5 being constant, this phenomenon will be 
more remarkable as sin2¢9is greater. Putting, therefore, sin 
20=1, we have gp=45°; whence it is evident that such a ring as 
we have been considering would be most conspicuous as a Ju- 
minous arch, after the setting of the cusp, provided it were placed 
at a distance from the earth’s centre =3956 v2, or 1640 miles 
above the surface. 
If, while we make sin 2e=1, we make also 2 sin 5=1, the 
ring will touch the horizon, and we shall have sin d=4= sin 30°. 
That is to say, the maximum depression of the sun at which the 
phenomenon can occur, is thirty degrees. 
e take the sun’s depression = 18°, as in the article referred 
to, then tangency will occur, when sin 29=2 sin 18°: which 
will give the value of e=19° 5’, or 70° 55’. If any value be 
assumed for ¢ between these limits, the phenomenon will be ob- 
servable; but for any value beyond them it will not be observa- 
ble at this depression of the sun. But the distance of the ring 
from the earth’s centre corresponding to any value of 9, is ex- 
pressed by the formula, 
peste 
cos 9 
in which D represents this distance, and R the earth’s radius. 
Whenee the limits of distance which will make the light visible 
at the close of twilight or at the commencement of the dawn, 
though the extremity may be beneath the horizon, are 4186 
miles for the lesser, and 12,100 miles for the greater, from the 
centre of the earth. 
The inclination of the ring to the horizon when it just touches 
the circle H/R’ is equal to 90°—ZR/=¢’. The geographical 
limits within which the light of a ring situated at any distance 
from the earth between the limits just determined, ought never 
to be absent in some form and during some portion of the night, 
