132 Algkbra. 



The law of coefficients holds good also. For : 



The coefficient of the first term is unity. 



The coefficient of the second term is ^^-{-a,,;^. Now P^ 

 stands for the sum of the n terms a^-\-a.2-\-a^-\- . . . a„. Hence 

 Pi-|-<2„^ 1, or the coefficient of the second term, is the sum of all 

 the different second terms of the binomial factors. 



The coefficient of the third term is V^-^-a^^^^V^. 



Now P2 represents the sum of all the different products of the 

 n letters a^, a^, a^, a^^, . . . a„ taken two at a time. 



(i). That is, P2 represents the sum of all the different products 

 of the n+i letters a^, a^, a.,^, . . . a,,ui taken two at a time 

 which do not contaiji a„_^^. 



Again, 



(2). That is, /2„^iP2 equals the sum of all the different prod- 

 ucts of the n-\-i letters a^, a^, a.^, a^, . . . a,^^^, taken two at a 

 time, which coyitain <2„| j. 



Therefore, putting (\) and (2) together, P2+«„. ^Pi equals 

 the sum of all the different products of the 71 -f i letters 

 «i, ^2> ^3> ^4. • • • <^^^^ 1, taken two at a time, both those which 

 do and those which do not contain ^,,+ J. 



Th^ coefficient of the fourth term is P3+a,,+ iP2. 



Now P3 equals the sum of all the different products of the 71 

 letters a^, a.^, a^, a^^, . . . a„, taken three at a time. 



(^3J. That is, Po equals the sum of all the different products of 

 the 7i-\-\ letters a^, a<^, ao^, a^, . . . «„|^, taken three at a time, 

 which do 7iot contain a„_^.^. 



Again, 



(4.). That is, «„4 1P2 equals the sum of all the different products 

 of the n-{-i letters a^, a^, a^, a^, . . . a,,^-^, taken three at a time, 

 which co7itain a„j^-^. 



Therefore, putting (2,) and (/\.) together, ^z~^^n\\^2 equals 

 the sum of all the different products of the ;/+i letters a^, a 2, 

 a^, a^, . . . ^„; I, taken three at a time. 



In like manner we may treat the coefficient of the fifth term, 

 and so on. The last term is the product of all the n-\-i letters 



«1, «2. «3> «4, • • • ««+l- 



