2(5 Mr Totter on the Aurora Borealis. 



of such a circle, depends on the possibility of the solution of this 

 question from the following data : A plane being tangent to a 

 o-iven sphere, given the angular bearings from the point of tan- 

 o-ence, of the points where a circle concentric with the sphere cuts 

 the tangent plane, and also the angular elevation of the highest 

 point of the circles to determine its radius ; the most moderate 

 knowledge of mathematics suffices to tell us that no such two cir- 

 cles can, on the same side of the pointof tangence, appear to coin- 

 cide exactly in all points. Let K A M, in Plate II. Figure 6 of 

 last Number, be the portion of a sphere to which the plane 

 Y A X is tangent at the point A, let the centre of the sphere 

 be the origin of the co-ordinates, let the plane of x % be in the 

 magnetic meridian, and the plane of xy parallel to the tangent 

 plane ; then CBD, being an arc of a great circle, we have 

 given the radius of the sphere (r,) the angular bearing of the 

 highest point of the arc B A X, which is in the plane of x z, 

 (let the trig. tang, of this angle equal e,) and the angle DA X 

 in the tangent plane, (of which let /'equal the trig, tang.,) let 

 the equation of the plane in which the great circle is, be x = u%, 

 then, if we have sufficient data to find the value of u, the cir- 

 cle is clearly determinate, but we have in the plane of x « the 

 following x = u *, ss = r + ex, and x" + xr - R 2 , where R 

 is the radius of the great circle ; and for the points where it 

 cuts the tangent plane, whose equation is %' — r, we have 

 x' — u r, y =/#', and a?' 2 -f y'~ + z" 2 = R 2 , or seven equations 



to determine seven unknown quantities, or eliminating m, *, x', 



2 

 «', z", and R, we arrive at this equation, u" — u~ — + u 



^lJ j J " — w ' or substituting for v its value — we 



e2 + e «p e + eP * r 



may put it under this form, x'"° — a/* r ■-■ + a! ^-^-^-l-r, — 



yo , which equations will, of course, produce threevalues, 



c-\-ef- 

 real or imaginary, for u and x. Now, on the afternoon of the 

 26th December last, the sky became very regularly covered at 

 distances with stratus clouds, and though some had very small 

 apparent altitude, yet the whole appeared to meet almost to- 

 gether in two opposite points of the horizon. If we take an 



