168 AN EXAMINATION OF SOME QUESTIONS 



The succeeding terms need not be written down, as the law is 

 obvious. In this, as in the former questions, the co-efficient 

 of « is p — q ; and if we employ the same abridgment of a 



for ( — 1)", &c. as in the last, the co-efficient of — -is 



ad -\- hd -\- cd 

 a e + be -j-2ce-j-Je 

 af -f 2bf+ 3cf + 2df + ef 

 ag + 2bg + 3cg -f 3dg + 2eg + fg 

 &c. &c. 



This series, which only differs from three times the sum of 

 all the products, two by two of the quantities a, b, c, &c. by 

 the omission of certain parts, may be called Q. On examining 

 what parts are omitted, it will be found that 



Gi-\-2a(^S(a-\-b + ..) — a — b — c)~{-bd 



+ b(^Sia + b + ..)—a — b — c^ 

 + (^ce + df+cg + ..) + 2(cd+de+ef+..) 

 + 3{ab+ac + bc) = 3S{ab + ..) 



— o {p — qf — ip + i) 



This equation becomes, by the mutual destruction of certain 

 terms 

 Q-f 2 a S (a -}- 6 + . .) + * 3 (a + 6 + . .) + («c + 6 rf + c c +..) 



+ 2{bc-{-cd + ..)-3 = 3 (r-^y-(P + '/) 



Hence 



