175 



will be constant or the same in all the differences of that order, 

 and will be = 



1.2.3.4 n ^■(;'— 1).(';— 2)...(^'— ;?— 1) 



l'^~iTj^'^~xi^^'~i^c. ].2.3...p^xl.2 3...rfle><1.2.3 ..e/x&c. 



For, if we expand the " powers of the polynomial quantities 

 in the above Lemma, by the multinomial theorem, 



In(P + Q + R + S + &c.)7 



we have P^~^ x Q^"*^ x R'^" ' x S'^-^x &c. 



with a co-efficient ^.(^— 1).(^— g).(^— 3)...(^-iP— 1) 



1.2.3...;»— f/x 1.2.3...(/—«x 1.2.3.. .e—/x&c. 



Alsoin (P-t-2Q + 3R + 4S + &c.)^ 



we have P'-x2 Q x3 R x4 Sx &c. , 

 with the same co-efficient. 

 Andin (P + 3Q + 6R+10S + &C.)? 



we have P?"^ x 3"" V x g'-'r'" x lO'-^S'-'^x &c. 

 with the same co-efficient, &c. 



th 

 But by the preceding proposition the ?j differences of 



1 X 1 X 1 X &c. 



2''-''x3'-%4*'--^x&c. 



3''""'x6'^~'xl0'~-^x&c. 

 &c. 

 will be constant and equal to 1 . 2 . 3 . . . . ?t 



p — d d — e e—f 



V xl.2 X 1.2.3 x&c. 



Therefore, 



