OPERATIVE DIVISION 397 



at length in " Verb Functions, "and it is unnecessary to do more 

 here than to give the results. 



Let us denote by (m) t the coefficient of the t th power of x 

 in the expansion of ( i -f- x) m , so that 



, s mini — i) . . . (m — t + i). 

 (m) t = -J '- jf '- 



Then if x n + px n - r =y, 



x = [o* + jxr-y s y, 



and [o n + ^o""]- 1 = oH — fczlVo' 1 ^- etc. 



L r — i\ n /j 



Put in terms of x and jy (which will be more familiar to the 

 reader), this gives 



I i Ir — i\ A -.J^? i (2r — i\ .. _»^I 



x = y* ( )Py » — ( )p l y » — 



^ r — i \ n /i 2r — i\ n / 2 



I- )py ■ —etc. 



Here w, r, £,>> may be positive, negative, integral, or fractional. 

 If r is positive, the series is the applied quotient of a descending 

 division ; if negative, of an ascending division. But of course 

 the series is not always convergent and not always real — matters 

 which will be examined later (Section V et seq.). 

 In the general case, if 



x n + bx n ~ l + cx n ~ 2 + dx u - 3 + . . . = y, 



then [o n + bo 11 ' 1 + . . -]" 1 = & ~~ bo" - etc., 



n 



and * = 3 k- y - {(L)c + (L)»y. - l{(^ + ©* + 



+ ©.^H- !{©.• + (& (aM + c%) + (^ b ' c + 



+ (n)/'K ? ' - H(»)/ + (n)* 2ibe+ Cd)+ 8>. 3 < W +bC,) + 

 + (^ + («) 5 fci K"- l{(n)> g + («)+ 2bf+ 2Ce + d ' ) + 



+ ^)/ 6W + i b ' e + c ') + {; t )S 6b ' c ' + 4b ' d ) + {i\s b ' c + 



+ @)5'WS-etc. 





