14 ON SHOOTING STARS. 



been smaller. Hence the mean distance of shooting stars from an observer is less than 232 

 kilometres, or 144 miles. 



Again, if tlie mean distance be computed by supposing the disappearing paths to bo 

 always those which are farthest from p, the result will evidently be smaller than the truth. 

 The approximate number of paths that disappear may be thus found. If none disappeared 

 the numbers in the last column of table II would be constant, and equal to 1800, the earth's 

 curvature being neglected. The quotients of 1800 divided by the numbers in that column 

 may be represented by q, and then q will express the proportion of all the paths along OA 

 that are visible at 0. We then should have the equation, 



l - a 2 sec 3 Op x* dx=: q I ~ a 2 see 3 p x 2 dx, 



where oo 1 represents the altitude of the farthest visible paths. Using summation and observing 



that 2' pa? = 8135325, we have 



« a 



8135325 q = S i /<.<-. 



a 



In this summation so many values of p x 2 as are contained entirely in 8135325^ are to be 

 taken, and with the next value of x such a value of p is to be found as will complete the 

 equation. Let p a stand for the values of p thus used, including the last. Then if d is the 

 mean distance from O of the visible paths in the cone, we shall have 



X 



sec 0-Vo a> 3 

 »= H 



2 l o -x 2 



■" Po ■*' 



a 



Computing 3 when is 5°, 15°, 25°, &c. , and the mean value of 8 for the whole heavens will 

 be equal to the expression, 



-~ .' , or 140.7 kilometres: 



- <P 



that is, the mean distance of all shooting stars from an observer (supposing the data on which 

 the computations are based to be correct) is greater than 140 kilometres, or 87 miles. This 

 limit, however, cannot be very positively asserted, since the errors from various sources, 

 especially those from using summation instead of integration, may in this case be quite con- 

 siderable. 



MEAN FORESHORTENING OF THE METEOR-PATHS BY PERSPECTIVE. 



To determine the effect of perspective in shortening the apparent paths of meteors, we 

 need the following geometrical proposition: 



If a sphere whoso radius is a bo supposed to have an indefinite number of diameters, and 

 the extremities of these diameters are uniformly distributed over the surface of the sphere, 

 and if O be a point without the sphere whose distance from the centre of the S23here is b, then 



will the mean value of the angles at O subtended by all the diameters be equal to „. j. 

 (304) 



