a (r) 



110 Prof. W. E. Story on Partial 



for any set of values of g u g 2 , g 3 , . . . ,g ic _ 1 that are not all 0. 

 Namely, it is evident that the constant terms of the us will 

 not occur in equations (43), being driven out by differentia- 

 tion ; they will be determined later. It may be remarked 

 that the notation (47) enables us to write equations (45) 

 thus : 



c = it G = 1 (50) 



The k — 1 equations (49) may be written 



-«£... =^pe.., r +l„ ('•= 1 - 2,3,..., K-land 1< G), (51) 



Jc=k-1 



whose sum, by (34), is 



1 IC = K — 1 



(«-!)« -(*-l)'^.,.-Q ^ l ^* +1 ). a ..»+l..' 



from which follows 



for any set of values of g^ g^ g 3 , . . . , g ic _ l that are not all 0. 

 Substituting oC u \_ from (52) in (51) we have 



(«-l)G *=i »^+i-- G- -*r+l» >( 53 ) 



(r=l,2,3,...,*-l) J 



for any set of values of g u g 2 , gz, • - > ■> g K _ x that are not 

 all 0. It will be observed that the aggregate multiplier of 

 C ..g r +l.. in ( 53 ) is 



g r +l . g r + l «-2g r +l 



(k-IJG" 1 " G ~ k-1 G * " 



The coefficients of the u's are all given in terms of the 

 coefficients of co by (52) and (53), excepting the constant 

 terms. 



We have, by (23), u r = In P r for x r = 1, that is, by (33), 



