88 OF PRESSURE AND EQUILIBRIUM. 



242. Definition. A series of units, a 

 series of natural numbers, a series of their 

 sums, and a series formed of the sums of all 

 the numbers of any preceding series, are 

 called figurate numbers of the first, second, 

 and other higher orders respectively. 



243. Lemma A. The figurate number, 

 of which the place is m, in the order n, is 

 pnnal in m(m + 1) (M+2) . . (m + n-2) 



equal to fTs . 3 . . (n-1) 



For, the two successive values of this expression, taken 



^ 1 J f (M— 1) M (M + 1) . . (M -f N— 3) 



for M — I and for m, are — V— ^^ ^ — ^^^r^ 



1.2.3. .(N — 1) 



, m(m + 1)(m + 2) ..(M + N-2) , . . ,.^. 



and — ^^ i — -—Ti ; — ^-r: —, and their difference 



1.2.3..(N — I) 



., ^ M(M + 1 )»(M + y— 3) _ M(M + 1)..(m + (N— 1)— 2 ) 



'^^^ ^' L2..(N— 1) i.2..(N— 2) ' 



which is the Mth figurate number of the order N — 1, 



.according to the definition: and when Nzz2, we obtain the 



1 . r-^^ c- 1 m(m + 1)..(m+n— 2) 

 natural series of ifitegers. Since also -4r-^ '- 



1.2.3...(M+N-2) -^ ■ u ■ .1. * 



= 1.2.3..(M-1).1.2..(N-1) ' '' " "''^""^ *^' '' ^"•^ ^ 

 are equally concerned in the expression, and the number 

 which occupies the place m in the order N is the same as 

 the number N of the order m. 



244. Lemma B. The binomial or rather 

 dinomial quantity {l+x) ^i-fNop+N--^ ^2 + 



N-l N-2 , , 



