7-PAKTITION8 01 143 



which is 



f 6.C 5 80c 4 2185.C 3 1361c 8 55899c 

 7.720 a+ 720~ 2 + 720 2 + 7.720 a 2.720 2 14.720 2+Ge 



=^0* f 6(^-l) 6 +135(c-iy+760(c-l)«-5046(c-l) ) . 



Let now ( 7 P X )' stand for the sum of the circulating terms 

 in x on the right of equation A; and ["«P,_i]' for the like 

 terras of ePj;-! . Then evidently 



( 7 P,)M,P^y=[.P^.i]'+[J^.J+ • • • +C.P-J'; (C) 

 for if we put in (A) x — 42 for x 9 and subtract the result from 

 (A), the remainder on the right must be the terms here pre- 

 sented. And it is plain that no power of x higher than the 

 third can have disappeared on the right of (C) by the sub- 

 traction. Let 



( 7 P,y=«^+^+c*+rf, ; (c) 



then (C) becomes 



o^— (*— 42) 8 )-fo(* 2 — x— 42) 2 )-fc(*— (>— 42)) = 

 -Lf | (-1350(^1 )*-5400(*-l))2 x _ 2 +(-1350(:r-8) 2 



—5400(a?— 8))2_ 

 + (— 1350(«— 15) a — 5400(«— 15))2_„+(— 1350(x— 22) 3 



— 5400(*— 22))2^3 

 + (— 1350(*— 29) 2 — 5400(#— 29))2 r _ ao - r -(— 1350(x—36)» 



— 5400(a>— 36))2.^ 

 +(*_ l)(+7926-6^!— 84246^ *— 16746_ ,+1176-6_ 4 



— 1674-6^ r-8424-6^,,) 

 + '— 8)(— 8424-6^,4- 7926-6^. 8 — 84246^ ^1674-6^,0 



+ 1176-6^— 16746^) 

 +(*— 15)(— 1674-6_ 19 — 8424-6^ ao +7926-6^ ir - 8424'6^ w 



--1674-6^ 7 +1176 6^_ 18 ) 

 +(*— 22)(+1176-6 - ^ M — 1674-6^ (r -8424-6 JP _ a7 +7926-6^_ a 



— 8424-6^ ar -1674-6^ M ) 

 +(*— 29)(— 1674 6^+1 176'6^— 1674-6^—8424 6^ 



+7926*6^*— 8424 6_*, 

 +(*_36)(— 8424 6^,— 1674 6,^+1176 -6^— 16746^ 



-842 If., u +7926-6^,)} 



