ON SHOOTING STARS. 15 



For let C be the centre of the sphere, AB a diameter, and the 

 angle ACO. The mean value of AOB will be twice that of AOC. The 



tangent of AOC is -. — -,. Let the whole number of the radii of 



° b—a cos o 



s 



the sphere be n, then the number which will make the angle ACO 



greater than 0, and less than + d0, is 2 n n sin Odd divided by 4 -, or \ n sin d 0. Multiply 

 the value of AOC for each value of by the number of radii for which ACO is between and 

 0-\-dO, and divide the sum of the products by the whole number of radii, or n, and we have 

 the mean value of AOC. This is evidently, 



TV Q/ 



The value of this expression, when b is greater than a, is j. j. Hence the mean value of 



— °n. 

 AOB is T . -=-, which was to be proved. • 



4 b * 



The mean effect of foreshortening by perspective may therefore be thus expressed. Let 

 the diameter of the sphere be bent into the arc of a circle whose radius is b, then the angle it 

 thus measures is to the mean value of AOB as a square to its inscribed circle. 



This result is independent of the ratio of a to b, except that it must be less than unity. 

 Hence the proposition applies to any number of equal, or unequal lines viewed by an observer, 

 provided only that the directions of the lines be properly distributed. 



If shooting stars came directly downward we should see all that were coming towards us, 

 since they would be near the zenith. We should see few, if any, whose paths are nearly at 

 right angles to the line of vision, for those woidd be down near the horizon, and concealed by 

 mists and smoke. It seems probable that in this case the mean effect of foreshortening would 

 be a little greater than for the diameters of a sphere. 



Again, if the paths were all parallel to the horizon we should see an undue number moving 

 nearly at right angles to the line of vision ; for those which are diminished most by fore- 

 shortening are near the horizon, and hence mainly hidden. 



But the directions of the meteor-paths are from all parts of the heavens, from horizon to 

 zenith. It seems reasonable to conclude that the mean effect of foreshortening is intermediate 

 between that for paths coming from points in the horizon and that for paths coming from the 

 zenith. Hence it ought to be nearly represented by that of the diameters of the sphere. 



This conclusion may be thus expressed. The mean length of the observed paths of 

 shooting stars is found to be 12°. G. If every path was turned about its middle point, and 

 bent into an arc of a circle of which the observer's eye was the centre, the mean apparent 

 length would be increased in the ratio of a square to its inscribed circle; that is ; it would bo 



equal to ^Xl2°.G, or about 1G°.04. 



(305) 



