OPERATIVE DIVISION 599 



TABLES 



Table I. Algebraic Multinomial Theorem. 



[o"']/« = (o + bo n ± l + co n ± 2 + . . .)"' = o" m + mbo mn ± 1 + {?nc + m^}o mn + 2 4- 

 + {md + m 2 2bc + >n 3 b 3 }o mn ± 3 + {me + m. 2 (c 2 + zbd) + m 33 b 2 c + w^ 4 }o"" l ± 4 + 

 + {mf+ vi.,z{cd + be) + m.^bc 3 + b'd) + m^b 3 c + w#}o" m ± 5 4- etc. 



Table II. Descending Inverts. 



i. If yjs^^x' 1 4- bx n ~ l + cx n -- + . . . =y = y n , then— 

 x = ^y = y - bin - {(i/n)c 4- (i/«)^ 2 } iy - {{*\n)d + {2\n),2bc + (2/n) 3 b 3 } J2y 3 - 



- { (3/»> + (3/«) 2 (^ + 2^0 + (3/«X3^ + ( l\nW } / 3y 3 - { (4/«)/+ (4/*)*2 (cd+ be) + 

 + (4/«) 3 3(^ 2 + b 3 d) + (4l»U^ + U/")^ 5 } /47 4 - {($\")g + (,S\»W 3 + 2ce + W) + 

 + (S!»W 3 + Med + ibh) + (5/«) 4 2( 3 ^V s + 2b*d) + (5/«) 5 5*V + (5W 6 } /5y 5 - 



- {{6\n)h + (6ln%2(de + cf) + (6/») 33 ^ 4- . . . (b = o)}/6y G - {{ 7 \n)i + 

 + {7JnUe 2 4- 2d/+2c/i) + (7/«)a3(^ 2 + ^0 + (7/«V 4 . • -}/7y 7 - {(8/«)/4- 

 + (Sln) 2 2(e/+ dg+ ch) + (8/n) 3 (d* + tcde + 3c 3 /) 4- (Sjn^^d . . . } /8y 8 - { {<)\n)k + 



+ (.9l"Uf 2 + 2e S + 2dh + 2 «") + (9/«) 3 3^ + C* + 2cd/+ c*g) + (9/«) 4 3^^ 4- 

 + ( 9 /«) 5 r i . . .} /9y 9 - {{io\n)l + (io/») 2 2(^ + ^ + di + cj) + (10! n) s3 (de 3 + d 3 /+ 

 + 2cef+2cdg+c 2 n) + (ioln) i 4(cd 3 + 3c 3 de)+(io/n) b $c t d. . .}/ioy 10 - etc. 



2. If -^ix = x 4- b 4- f^" 1 4- <£r 2 + <?-*~ 3 + . . . = y = y, then— 



x = ^y = y-b-c/y- {d + be} /y 3 - {e + (c 3 + 2bd) + b 3 c}\y 3 - {/+ 3 (cd + be) + 

 + sibc 2 + &d) 4- b 3 c}\y k - {g 4- 2(d* + 2ce + 2b/) + 2^ 4- tbed 4- &e) 4- 

 + 2(3^ + 2b*d) + b*c}y s - {h 4- ${de 4- c/+ bg) + \o{c 2 d 4- bd 3 + 2bce 4- b 3 /) + 

 4- loibc 3 4- iPcd 4- b 3 e) + ^bh 2 4- b 4 d 4- <5V} /y 6 - etc. 



3. If ^x^x 3 4- bx + c 4- <&r -1 4- &r~ 2 + . . . = y — y'\ then— 



.r = ^.] x y =y-b/2-{2 3 c- b 3 }J2'y - dJ2y 2 - {2 6 e + 2*{c 3 + 2bd) - 2 3 b 2 c 4- b 1 } /2 7 y 3 - 



- {2 9 ^ 4- 2 7 . 3(rtT 3 4- 2ce 4- 2^/) 4- 2 6 (^ + 6^/ 4- 3^) - 2 4 (3^V + 2b 3 d) + 

 + i2b*c - ^ 6 }/2 , V - {h 4- 2(<fe 4- ^/4- bg) 4- {fd + bd 3 4- 2*« 4- ^/)}/2y 6 - etc. 



4. If y^sX ~ x 3 + bx 3 + ex + d + ex- 1 + . . . =j/ = y 3 , then— 



^=^>=y- ^/3- {3^-^}/3V- {3 3 ^- 3 2 ^4- 2b 3 l !3V -e/3y<- {3 5 /+ 3 4 (^4- ^)- 



- f(bc 2 + ^V) + 1 5^V - 2<5 5 }/3V - {3 7 £ 4- 3 6 (^ 2 + 2^ + 2<J/) - 3V s 4-6<5*/4- 3^) + 

 + 3 S (6^V 8 + 4^*0 - 3 2 . 7 ^V 4- 7^ 6 } /3 8 y 5 - {^ + (<& 4- ^/+ ^)} /2y 6 - etc. 



5. If ylf r v = x* + bx 3 + . . . = y = y 4 and ^s* = -r 5 4- &r* 4- . . . =y = y 5 , then — 

 x = y - b/4 - {4 3 c - 6b 3 } /4 3 y - {A 3 d- 4 . zbc 4- 2^} /4V - {4 5 ^ - 4 3 . 2(^ 2 4- 2bd) + 



+ 4 2 . 10^4- 3o£ 4 } /4V "//4y 4 - {4V+ 4 5 - 2(^ 3 4- 2ce 4- zbf) - 4 4 . 2(^ 3 4- 6bcd+ $b 3 e) + 

 4- 4 3 . 7(3^' 4- 2Pd) 4-4.7. 22^V 4- 77b 6 } /4V ~ etc - 

 x = y - b/s - { 5^ - 2£'} / 5 2 y - { 5W - 1 5^ 4- 4^ 3 }/5 V - { 5 3 ^ - 5 2 (^ 8 4- ibd) 4- 5 . 7^ 



- 7b'} IsY- { 5Y- 5 W4- be) 4- 5 s ■ 3(^ 4- Vd)-n&c + 44^ 5 j /5V -^/5y 5 - etc. 



