Prof. Encke on Interpolation. 35 



sions be called difference-quantities^ and denoted according to 

 the quantities which are used for forming them by \j) . g], \jp.q ■ »] 

 &c. The formulae being symmetrical, {jp.q- ?] and [</ . r.jp\ are 

 identical, or the letters in these expressions may be arbitrarily 

 exchanged one for the other. 



By the addition of the different values, the formula (I) as- 

 sumes the following form, which is more convenient for use : 

 X4=P + {oc-p) [jp.?] + {x-p){x-q)lTp.q.r-\ 



A-{x-'p) {x-q){x-r)l'p.q.r.s'\ (11) 



The formation of the above difference-quantities will be 

 most easily perceived by subtracting from each other two such 

 quantities of the same dimensions containing the same ele- 

 ments but one. Thus, for example, [? • ?' • sj— [j^ . 5' . r] = 



S o 5 1 2 ^ a-Q\ ^- 



{s-q) (s-r) "^ ^ < (r-y) (/-.) (»-;,) (r-q) S'^ ^X {q-r) {q-s) 



I ) _p 1 s R(5-;') 



(?-/')(?-'•)> {p-q) ip-r) is-q)(s-r) (r-;))(r-y)(r-j) 



and it will be easily seen that generally 



lq....yz]- [;?.... 3/] =(2-p)[p..-.^2] 



If we conceive, therefore, these quantities, to which, for 

 symmetry's sake, the quantities P, Q, R, S may be reckoned to 

 be placed in the following manner, — 



P 



J-] [p-g^-r-s'] 



s.tl 



[q.r.s.t] 



[p.q.r.s.t] 



each vertical column will be formed by subtracting a term of 

 the preceding vertical column from the one below or the same 

 column, and by dividing this difference by the difference of the 

 arguments, to which diagonals drawn through the next higher 

 and next lower quantities will point. For we have 



[p-Ql = 

 [p.q.r] = 



[p.q.r.s] = 



q-p 



■ '•] - [;>• ?] 



r — p 



'• ■ s]-[p-q-r] 



s-p 



In tlms applying tlie formula (II) it will be always noces- 

 F 2 sary 



