Mr Potter an the Aurora Borealis. 27 



example and suppose that a cloud, or an arch of an Aurora, 

 which was a portion of a great circle, for an elevation of 30" 

 in its centre, extended 170° on the horizon, we find by the 

 above formula that such an object, taking the smallest value 

 of u which the formula gives, must be at rather more than 20 

 miles in height to fulfil the conditions; but the arches of the 

 aurora are never observed to have so flattened a form as this; 

 and when we apply the formula to actual observations, the 

 height comes out so great as to be quite improbable. And, 

 speaking from recollection only, I believe that two arches which 

 I have "seen had so much apparent curvature, or had so little 

 extent on the horizon, compared with their altitude, that when 

 I attempted calculations for them, on the supposition of their 

 being arcs of great circles, the three values for u came out 

 quite imaginary; and such as were impossible to be constructed 

 o-eometrically. Hence this supposition must be abandoned. 



From all we know of the magnetic variation at present, it is 

 more probable that Mr Dalton's supposition, {Meteorological 

 Essays, page 181,) that the arches are similar to parallels of 

 latitude around the magnetic pole, is much nearer the truth 

 than the former supposition. On this ground the problem is 

 equally determinate with the other, or we may say more so, as 

 we shall find that only one such small circle can fulfil the con- 

 ditions required from the data ; whereas in the other, there 

 are, real or imaginary, three values for R. It will be easily 

 seen by any one acquainted with analytical geometry, on re- 

 ferring to Figure 7, that in this case we have the following 

 equations, * = r + e w, aP + ^ = R s in the plane of a, * and 

 y' =faf, ad- + if + z'~ = R 2 in the tangent plane, having 

 z' = r for its equation. And also, as we consider the arch 

 a portion of a small circle in a plane perpendicular to the sup- 

 posed magnetic axis, we have {x — af + (* — c)» = R' 2 , 

 and {af — a? + tf 1 + (** — cf = R'-, taking R', the radius 

 of the small circle considered in its own plane, where we have 

 nine unknown quantities, and seven equations ; but if wc 

 take g equal the trig. tang, of the complement of the mag- 

 netic polar distance of the place of observation A, or of 

 the angle, M O .r, we establish the following conditions, 

 c = ag and a- + tfl + R* = R 1 and the whole of the un- 



