4 oo SCIENCE PROGRESS 



H. Prove that (0: ) 1 ( )" r =(^; 1 ) r 



Put — n and — r for n and r in either of the series just given 

 and we shall obtain the other. 



Hence <f>:\ . fa 1 = i . 



Obviously from the definitions of <p a and yjr n in the previous 

 subsection 



tyn= [^-^o- 1 and (/>„. = [-^_„]o-'. 



Hence +? = [[^jo" 1 ]" 1 = [cr^l* == -^ ; 



with a similar identity for <\)~ l ; and these identities are now 

 seen to verify the series for <£ _1 and ^r-\ (Note that [o -1 ] -1 = o _1 .) 

 This proposition enables us to reduce ascending to descend- 

 ing division and vice versa. The same results are obtained by 



putting x — -in numerical equations ; and the reader should 



study the details . It will be found that the two forms of division 

 if taken according to the equation give the same series as a result. 



I. Prove that -frf- 1 = o, ^,; 1 ^ = o, and O^ 1 ]" 1 = o ; 



with similar identities for <f) n . For each term of the superior 

 operation substitute the proper power of the inferior operation 

 as expanded in Example G, and collect the coefficients of the 

 same powers of o in the result. The issue will be found to be 

 0=0— thus proving that we are dealing only with absolute 

 identities. But all the terms must be considered. 



J. Prove that the invert by descending division of an 

 equation which possesses only positive powers of the variable 

 contains all its real and imaginary roots. Take for example 

 x" + px n ~ r = y, which has n roots. Then by (2) above, its 

 invert is a function of %/y, which also has n roots. Let these be 

 ma, ma?, ma? . . . ; so that, by a well-known theorem, 



a + a- + a 3 . . . -f a" = 0. 

 Insert these values of </y successively in the series for x, that 

 is, in [tK ]~'.r. We thus obtain n different series. It will be 

 found that the sum of these series and the sum of their products 

 two, three, four ... at a time all vanish, until we come to the 

 sum of their products rata time, which = ± p. The sums of 

 their products more than r times together also vanish, until we 

 come to the product of all the series together, which = ± y- 

 This satisfies the familiar law that the sums of the products of 



