Gruson's Enlwickelung von jc« + y in einer "Reihe. 3 1 



8) x8 + y» =(x + y)[x' + y'] — xy[x«-f y«] 



= (x + 7)' — Oxy(x + y)»-!- 2ox'y* (x + y)*— iCx'y» (x + y)* 

 + ax+y* 



9) x'+y» =(x + y)LX«+y^]-xy[x'+yn 



= (" + y)'— 9xy (x4- y)' 4- 27X*y* (x+ y)^— 30X V^x-f- y>3 



+ gx-'yCx + y) 



|o)x»» + y»°=(x+y)[x»+y«'] — xy[x8+'yS] 



= (x + y) ' °- 10 xy (x+y)»+ 35X >= (x+y)* — 50X»y' (x+y)* 



+ 25x*y*(x + y)* — 2X*y*^ 



M}x»»-f y" = (x + y)[x'o + y'o]—xy[x*'+y*] 



= (x+y)" — Iixy(x+y)«+44x'y'(x+y)'— 77x»y'(x+y)s 



+ SSx^y^Cx + y)' — nx*y*(x + y) 



12)x'*+y'*=(x + y)[x»«+y»*] — xy[x*° + y'°] 



= (3'+y)'*— I2xy(x+y)'°+54x»y*(x+y)'— Il2x3y3(x+y)« 



+ losx-'y^Cx+y)^— 5Cx^y*(x+y)* + 2x«y'' 

 etc. 



Diese Entwickelungen berechtigen schon, zu schliefsen, dafs die allge- 

 meine Form sein wird 



x« -Uyn— (x + y:» -f AjxyCx + y)»-» -f AjX^y'Cx + y)"-* 



-|- A3 X» y3(x + y)»— 6 + ... -f Ar x'y' (x+y)"— ^^^ + • ■ • 

 also . ,, •• . ; /)..;■' . ' ;■ .- ' -1. 



etc. 



llieraiis ergehen sich folgende Gleichungen 



•2( + A,=o 



"18 + ""'?f-A,+A, =zo- 



»e + "-Q3. A, 4- "-^Sf A, +43 .;= o 



■'!D + "-'€. A, +'--:B,A,-f r'^Sf.Aj + A, ^ o-:.,,^ , j 



"g 4- "-©. A, + "-^. A, + "-^23. A3 + »-8'X A^ + A j = o 



"iK-f- "-'JXiA, +"-*9i.A, +"-«9{'. A3 +°m A, -f""'°^.Aj + . . . 



-f»-^"-'>9r.A,_.H-A, = o 



