436 Prof. Sylvester on the Integral of 



case, of internal resistance. However this may be, the foregoing 

 experiment enables ns to draw some important inferences. 



Storms at great heights must be greatly facilitated by the 

 mobility of the particles of the air. In fact storms are cases of 

 convection on a large scale, and in our front chamber we had one 

 in miniature. With the same difference of temperature on the 

 summit of Mont Blanc, the motion of convection would be very 

 nearly twice as great as at the sea-level. 



In the summer of 1859 I was fortunate enough to induce my 

 friend Professor Frankland to accompany me to the summit of 

 Mont Blanc, and to determine the comparative rates of combus- 

 tion there and in the valley of Chamouni. Six candles were 

 purchased, burnt for an hour at Chamouni, and the loss of weight 

 determined. The same candles were lighted for the same time 

 on the summit of the mountain, and the consumption deter- 

 mined. Within the limits of error, the consumption above was 

 equal to that below. The light below was immensely greater 

 than that above, still the amount of stearine consumed in the 

 two cases was sensibly the same. Professor Frankland surmised 

 this to be due to the greater mobility of the rarefied air, which 

 allowed a freer interpenetration of the flame by the oxygen* ; and 

 the foregoing experiments show that the augmentation of mobi- 

 lity is just such as would account for the observed effect. 



LIX. On the Integral of the general Equation in Differences. 

 By J. J. Sylvester, A.M., F.R.S.-f 



THE most general form which can be given to a linear equa- 

 tion in differences may easily be seen to be reducible to 

 the following, 



a x u x + b x u x _i + c x . u x - 2 + &c. ad lib. = 0, 

 with the initial conditions 



Consequently to find u n) or let us rather say to find 



^— j wj wg • • • a n Ufi) 



is really the problem of finding the value of a determinant belong- 

 ing to a matrix of n* terms, whereof all the places below the 



* The influence of interpenetration is well seen in the exposed gas-jets of 

 London, particularly in the butcher's shops on a Saturday night. A gust of 

 wind, which carries oxygen to the centre of a flame, suddenly deprives it of 

 light. A simple and beautiful experiment consists of passing a lighted 

 candle swiftly to and fro through the air ; the white light reduces itself to 

 a pale-blue band. Bunsen's burner is an illustration in the same line. 



t Communicated by the Author, having been read at the Meeting of the 

 British Association at Cambridge, October 1862. 



