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



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



3 



ad -f bd + cd 

 ae -f be -f 2ce -f de 

 af + 26/+ 3c/ + 2rf/ + <?/ 

 ag + 2bg + 3cg + Sdg -f- 2e# +/g 

 &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 



Q-f 2a(S(a + 6+..) — a — b — c)+bd 



+ 6 (S (a + 6 + . .) — a — b — c) 



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

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



— o (p — ?) 2 — (p + q) 



— <* g 



This equation becomes, by the mutual destruction of certain 

 terms 



QH- 2 a S (a + b + . .) + b 3 (a + 6 + . .) + ( ac + * d + c c +..) 



+ 2 (6c + cd + ..) — 3 = 3 fr-^-fp + g) 



Hence 



