Mr Potter on the Aurora Borcalls. 215 



Then it will be seen that the arch being at an equal height 

 irt every part above the surface of the earth, it will be every- 

 where equally distant from the centre, * we will represent this 

 distance O e by R, putting r for the mean radius of the earth, 

 and r' that of the small circle considered in its own plane. We 

 have then for the points of the small circle in the plane of x z, 

 z — r + ex, x* + & = R 2 , and {x — af + (z — c) 2 = r' 2 , 

 and in the plane daf, z" — r +jx', y' —j'ix', x'~ + y n -j- z? 2 

 = R 2 , (<*?' — a)°- + y n + {%' — c) 2 = ) ,/2 , we have also c= ag 

 and R- = r* + a? + c 2 . 



These equations are sufficient to determine the circle. In 

 the elimination it is better for simplifying the formula to put 

 I = 1 + i 2 / 2 + f, m = 1 + e\ n = 1 + g J , o = 1 + jg, 

 and p = 1 e + g. 



We then arrive at this equation, 



(/ p" g + 2 e o- p — ?no? g — 2 / o p 2 \ ... 

 r-rr 5 s ~ — — I which we see is 

 I p- n — m o z n / 



of the same form as the one we found in considering only 

 the points of intersection on the horizon ; and, in fact, if 

 we make the angle of inclination equal to zero, it then co- 

 incides with the horizon; and we have i = radius = 1, and 

 j = o, these give us / = 1 -\-j~ and o — 1, and the equation 



i i / lp* g + 2 ep — m g\ . 



above becomesa = r I - ' *-. \, 1 the same as we 



\ lp- n — m n / 



found before. As the height of the meteor, and its distance 

 from the observer, are generally the only elements required to 

 be known, we may shorten the calculation by finding the imme- 

 diate analytic expressions for them. Thus, the height being 

 R — r, we have onlv to find the value of R and subtract from 

 it the radius of the earth ; but. substituting the value of a in 

 the above equations, we find for the horizontal plane the follow- 

 ing easy and short formula, viz. R = 2 r */( fj^f^-^vi + z) 



• We here neglect the spheroidal figure of the earth, as it remains yet 

 to be determined whether the observations can be taken so accurately, on 

 such a meteor as the aurora, as to render such nicety in calculation of'ser- 

 vic. If it is f'ouiul that they can be, it will become necessary also to cor- 

 rect the angles observed, from the consideration, that the arch is a view of 

 a cylindrical ring, of which the figure of the section would require to be 

 rtained. 



