OF PRESSURE AND EQUILIBRIUM. 89 



By actual multiplication, we find 



1+07 



'i-{-2x-\-x^ 

 xxzz x + 2x"-\-x^ 



l-\-Sx-\Sx'^ +x^, and the coefficients are 



(N) 



1,1 each being obtained by adding together 



1,2, 1 two contiguous coefficients of the pre- 



1.3, 3, 1 ceding lines ; whence it follows, that each 



1.4, 6, 4, 1 of the vertical columns must contain a 

 1,5,10,10,5,1 series of figu rate numbers of an order 

 1,6,15,20,15,6,1 indicated by its distance from the begin- 

 ning, the place of the coefficients in the 



order being lowered by one at each step, so that for any 

 horizontal line answering to the power N, we have I, Ng, 

 (n— 1)3, (n— 2)4 ... denoting the place of the figurate 

 number by the letters N, (N-— 1). . , and the order by the 

 figures below. Now, the third coefficient, (n— 1)3, put- 

 ting 3 for " n", is —^Yo — f ^°^ ^^®° substituting N — I 



ffif lw 



for M, ■ — : in the same manner the fourth coefficient 



/.T ON M(m + 1)(M+2) . (N-2)(N-1)N 



(N-2)4, or -^^ — j-^^ ^, becomes -^^ ^^ — '-— ; 



and the subsequent terms may be shown in a similar man- 

 ner to follow the same law. 



Scholium. This demonstration is only strictly appli- 

 cable to integral and positive powers, such as are very 

 properly denoted, in the article Fluents of the Supple- 

 ment of the Encyclopaedia Britannica, by small Roman 

 capitals : it may be extended without much difficulty to 



