Intelligence and Miscellaneous Articles. 463 



Ainsi, par leur comparaison avec 1'expression 

 cos u 4- sin u s/ — 1 , 



les imaginaires ont servia determiner les quantit&s replies; et l'on en 

 voit des applications nombreuses dans le calcul infinitesimal. 



Supposons 2° que h, abstraction faite 8, gne, soit > 1 . 



Dans ce cas, on fera r 3 = tang m, 

 ce qui donne 



d'ou Ton tire 



3 _L 2 



sin 2 u = — . 



h 



Par une operation semblable a la pr^cedente, on d£terminera les 

 trois valeurs de x, dont une est r^elle, et les deux autres imaginaires, 

 savoir 



•v/f. 



* = r. , 



2 tang u 



(u' etant = v tang u) j 



et x = - x /f . /cos2«'±V~3Y 



V 3 \ 81112 m' / 



Fayolle. 



LUMINOUS BELT OF SEPT. 29TH. 



About ten minutes or a quarter past eight o'clock in the evening 

 on Monday 29th September last, I observed an uncommon luminous 

 arch, which I imagine was the Aurora Borealis, having a different 

 appearance to what is usual. This arch extended from west to east j 

 I cannot say that it reached from one part of the horizon to the other 

 completely j the western part of the Aurora appeared bent towards 

 the north, and like a straight column of light, which column ex- 

 tended upward beyond the lower portion of the bow towards the 

 south. When I first (or soon after) saw it, the vertex of the arch or 

 bow appeared to include part of the constellation Ursa minor ; the 

 stars p and y, I think, were included. Some time afterwards the ver- 

 tex was rather southward of the zenith : the apparent change of place 

 of this phenomenon was so rapid that 1 supposed if I stood still I 

 might see the motion j but in this opinion I was disappointed. Du- 

 ring my observing this appearance, 1 walked some distance from Hale- 

 End towards Woodford and back : this change of place I imagine 

 made no difference in the apparent situation of the bow, with respect 

 to the stars j as the Aurora was probably at too great a distance to 

 have any effect in this respect. After the arch was gone, there re- 

 mained in the west the straight leg of it (if so it may be called) for a 

 considerable time, and then the appearance was much like the sun- 

 beams 



