Prof. Eiicke on Litcrpolation. 91 



combustion applied to the nine pounds of steam or water ge- 

 nerated ; and therefore its temperature considered as water, 

 would be i-aised as much more than 180° as 410 exceeds 

 9 (l^il X 180), or 8200 of Fahrenheit's scale. If we knew with 

 more accuracy the proportionate capacities of water and of 

 these gases, we might thus tolei'ably ascertain the actual ratio 

 between that caloric which is chemically combined and which 

 seems to constitute the gaseous state, that which is lost by 

 temperature, and that which is lost by reduced capacity. If 

 we assume that, when these two gases are reduced to the 

 contact of particles by a temperature of —448, they have the 

 same capacity for caloric as water, and that in the condensation 

 they have lost 1000 degrees by reduced capacity, we shall have 

 the proportion of that which is moveable by both causes, when 

 compared with that chemically combined and permanent, as 

 1448 to 8200, or nearly as 1 to 5-7. It is of great importance 

 to ascertain the actual commencement of our scale of tempe^ 

 rature, as it would lead to an accurate knowledge of what cer- 

 tain determinate quantities of caloric can effect, and it would 

 come measureable, as the other constituents of compound bodies 

 are. It is very probable that caloric will be found to combine 

 in distinct doses when it enters into those chemical unions not 

 affecting temperature. If we assume — 448 as the true zero, 

 and assume 20° as a dose of caloric, the history of water in its 

 relation to it would stand thus : We should have twenty-four 

 doses or 480° of heat in ice at 32°; an addition of seven doses, 

 or 140°, melts it to water; nine doses, or 180°, brings it to the 

 boiling point under atmospheric pressure, and forty-eight 

 doses more, or 960°, generates steam at 212°, 



I am. Gentlemen, yours, &c. 



Geo. Cayley. 



XIII. On Intei-polation. 



{From Prof. Encke's Astronoin. Jahrhuch for 1830, p. 265.) 



[Concluded from p. 36.] 



"POR calculation it is most convenient to correct successively 



-*- each of the difference-quantities by the following one, and 



separating the common factors to write the formula as follows : 



(VI) X = A + (x-a) {[«i] +ji--6{[«.«i] 



The factors are here used in the following order : 



(•^-^„)' (^-«„)» (^-^,_i)> x-a^^_^...{x-a,\{x-h\{x-a). 



If we proceed, therefore, from ^r, and then first take the 

 N 2 nearest 



