39 8 SCIENCE PROGRESS 



The values of the coefficients are not restricted. The 

 number of them is unlimited ; but if it exceeds n the original 

 equation contains negative powers of x — as happens when we 

 employ the various settings suggested in the last subsection. 



It is easy to construct further terms of the series. If in- 

 stead of b, c, d, . . . , we use the weighted coefficients pi,p 2 ,p3, • ■ ., 

 we shall see that for the groups contained within the large 

 brackets the sum of the products of weight and power is the 

 same for all. Thus if we write p x % pz for the group b*d occurring 

 in the last major term given above, we obtain for it the number 

 1x3 + 3x1 = 6 ; and we shall find that the weight of all the 



6 



other groups affected by y~« is 6 also. Moreover, the expressions 

 contained within the small brackets are each evidently made 

 up of all possible groups of the first six coefficients p x to p e 

 which have the same weight and are of the same order. Lastly, 

 the sum of the numerical coefficients of the groups within the 

 small brackets is an appropriate binomial coefficient — it is 

 (S) r in the case of the last major term written out. In short, the 

 major coefficients of the invert are nothing but the multinomial 

 coefficients multiplied by a numerical factor, and are therefore 

 well known. The law is as follows. Let 



(1 + bx -f ex" + dx z . . .)'" = 1 + {m} x x + {m} 2 x + {m} 3 x + 



. . . [m} r x + . . . ; 



and <t> n = 0" + 60" + ' + co n+a + . . ., 



and -f „ = o n + 60" - 1 + co"- 2 + . . . ; 



then [</,„]" ! = 0* + -I - -1 o« + I [ - 4 0! +- j - i[ o= + . . . 



r, -1 1 i ifol -° 1 fO - 1 - * i 2 \ n -~ ■ 

 Lr J o[«Ji 1 (wjs 2{nJ3 



It is shown in " Verb Functions," page 51, that these results 

 depend upon the property of the multinomial coefficients which 

 is contained in the equation 



{m} r -i + -{ - 2m} 1 {2m}r-2 + \{ - 3w} 2 {3w}r-s+ 



. . . + { — rm} r -\ = o, 

 r 



and which holds equally well, of course, for the binomial 

 coefficients. 



