442 A Simple Demonstration of a remarkable Proposition 



l h 45 m . The phenomenon had finally disappeared, and the 

 haze in which these halos were formed had passed away and 

 the sun again shone brightly. 



From 2 h till 3 h a rain-cloud passed along the south-east 

 horizon, but it never rose higher than 15°. 



At 4 h much cumuli passed over together with some scud, 

 and a few drops of rain fell. In a few minutes the sky was 

 clear with the exception of iew cumuli. Starlight night until 

 ll h , then overcast. A near resemblance to this phenomenon 

 was noticed at Derby on the 16th of June 1843. (See page 32 

 of Lowe's Atmospheric Phenomena.) 



The 1 8th had been a thorough rainy day, and l'fVuo" mcn 

 fell ; the wind was from the south-east, with the barometer 

 stationary at 29*296 inches* until 8 h when it began to rise; 

 at 9 h it had reached 29*307 inches ; it continued to rise all 

 night, and at 21 h of the 1 9th was 29*426 inches. At h it had 

 fallen to 29*424 inches ; it remained then stationary until 8 h . 

 At 9 h it had again fallen to 29*401 inches. 



The wind on the 19th was south-west all day, and the mi- 

 nimum temperature (seven feet from the ground, N.N.E. 

 aspect) was 48°; on grass 41°, the maximum temperature 

 (seven feet N.N.E.) 58°*5. The mean temperature during 

 the time the phenomenon was visible was 57 0, 5, and the 

 hygrometer 53°*5. The halos were all of the same width, 

 viz. 1°. 



This phenomenon was formed in a haze decreasing in den- 

 sity: line weather therefore followed this appearance. 



LXIII. A Simple Demonstration of a remarkable Proposition 

 relative to Polyhedrons f . 



THE proposition expresses the relation between the num- 

 ber of faces, the number of corners, and the number of 

 edges of a polyhedron. 



It was first demonstrated by Euler in the Transactions of 

 the Petersburg Society for the year 1758. Legendre has 

 given a demonstration of it ill his Elements of Geometry 

 (English translation, p. 180). His demonstration is quite 

 satisfactory, but it is founded on the rule for finding the area 

 of a spherical polygon, which is not very easily reached ; 

 whereas the demonstration which is now submitted to the 

 reader, rests entirely on the definition of a polyhedron, and 

 can be followed with the greatest ease. 



* The barometric observations are corrected and reduced to 32°. 

 f Communicated by the Author. 



