1875.| The Atmospheres of the Planets. 461 
where— 
6” =sin-* (sin 0+d) 
§'=sin-"(sin 0—d) 
d being the radius of the shadow. The breadth of the 
shadow will be obviously equal to 2d. Now, considering 
the mean radius of the shadow to be half a second of arc 
under these conditions when 0=9° (for its maximum value 
11° it will be much greater), the ratio of the equatorial dia- 
meter of the satellite to the polar will be, when the centre 
of the shadow is in the following apparent ecliptical longi- 
tude,— 
+80°=1'go —75 =0°40 
+ 60°=1°27 —60°=o0'°71 
gt E545) 0704 
+30 =1°07 —30 =0°g0 
The apparent centre of Jupiter being in +9’, of course all 
the positive longitude must be diminished by 9° and the 
negative increased by 9° to obtain the position relative to the 
apparent centre of the disc. It is evident, therefore, that 
for nearly two-thirds of its course after entering the disc 
before reaching the centre the shadow will appear as an 
ellipse, the elongation gradually diminishing. Near the 
centre it will be circular, and for about one-third of the 
final half of its passage elliptical in the reverse direction. 
_ These variations have been detected with a six-inch equa- 
torial during this year; though the equatorial penumbral 
fringe which must necessarily be present has not been dis- 
tinctly seen, probably on account of its exceeding delicacy. 
Comparing the effect of this cause with the only six cases 
mentioned definitely in the paper referred to,* it appears 
that in four of these the ellipticity arising from this case 
must have been sufficient to render it distin¢tly visible ; in one 
case the shadow was round, and in the other most marked'y 
elliptical in the reverse direction to that seen. As under 
any conditions these variations must be undergone, the effect 
that they would have in combination with the hypotheses 
assumed, would be to make the dark shadow to all intents 
vanish on occasions before reaching the limb of Jupiter—a 
fact in itself sufficient to condemn the hypotheses. But 
other causes might readily produce the effects noticed, espe- 
cially the clouding of the atmosphere of the satellites, 
whilst the extraordinary irregularities undergone by the 
satellites themselves show the uncertainty that must attach 
to such extremely delicate observations. 
* M.N., vol. xxxv., p. 65. 
