578 



SCIENCE 



[N. S. Vol. XLII. No. 1086 



Hence we have 

 (13) 



2 0,uS 





V"i: 



Let us now suppose that two positive quan- 

 tities a and s, and a positive quantity P < 1, 

 exist, such that 



a(l-p)=Oiiga(l+p) (i = p+l,p+2, ••■,?»), 

 (14) a(l-p)^02.-^a(l+p) (i = l, 2, •••,m-e), 



s(l-p)Ssi <s(l+p) (i=l,2, •■■,m). 

 It follows readily from (13) and (14) that 

 m — p — q 



(15) 



(l-rf)' 



<{n 



- p) (to - q) 



\i -p^ • 



< R 



V(m — p) (to — g) 

 Similarly from (4) and (14) we have 



(16) 



(n^)' 



■V(m — p) (to — g) 



<(B-:)': 



V(m — p) (m — g) 



We might obtain still narrower limits for 

 the values of R and r than those given in (15) 

 and (16). It is apparent from the limits ob- 

 tained, however, that if P is sufficiently small, 

 r will furnish a good approximation to the 

 value of R. 



We will now consider the case where some 

 of the a's on the right-hand side of the equa- 

 tions in (3) are negative. Let us suppose that 

 the first A of the (m — p — g) tj's that appear 

 in both equations have coefficients of the same 

 sign in the two equations, and that the re- 

 mainder, /}. in number, have coefficients of 

 opposite signs. Obviously, an increase in x 

 that is uniformly distributed with regard to 

 the e's involved in x, will be accompanied by a 

 decrease in those e's for which the correspond- 

 ing -qs have negative coefficients in the first 

 equation in (3) ; also an increase in an yj 

 having a negative coefficient in the second 

 equation will cause a corresponding decrease 

 in the value of y. Hence we have for the 

 ratio in (9) 



(17) 



»=p+A tt=m — q 



2 K»|s„- 2 I 



2 1 Qlr I S, 



a/;, 



2 a,u^si 



V 2 Ot^^Sv 



Let us now suppose that the a's and the s's 

 satisfy equations of the type (14), i. e., equa- 

 tions obtained by replacing the a's in those 

 equations by their absolute values. Then it is 

 easy to see that if A > /*, and P is sufficiently 

 small, 



\-u. (1 -I- p=) (1 - pY 



-^{m — p)(rn — q) 



2{\ + p) 



(1 + pY 



p(l - pY 



< R 



(18) 



■•i{m~p){m- q) (1+p)^ 



X-M (l+p2)(l-t-p)2 



■>i{m-p){m- q) (1 - p)^ 



^ 2(X-|-m) _ pO-+p^) 



<{m — p){m-q) (1 - p)* 



Furthermore, in view of (4), we have for r 

 X-jH 1 4- 6p^ + p^ 



<{m -p)(m- q) (1 + p)' 



4(X-|-m) p(l+p=)^_ 



(19) 



V(to - p){m- q) (1 + P)* 

 X - M 1 -!- 6p2 - 



^(m — p){m — q) (1 ~ p^ 



^ 4(X + m) . pd + p") 



V(to -p){m - q) (1 - Py 



The corresponding inequalities for the cases 

 where \%ix. are easily obtained. It follows 

 from (18) and (19) or the corresponding in- 

 equalities, that r will be a good approximation 

 to i2 if P is sufficiently small. 



The case where all the a's on the right-hand 

 side of (3) that are not zero, are negative, 

 does not seem to have any great interest in 

 connection with the applications discussed in 

 this paper. In any event the treatment of that 

 case presents no new difficulties, so we shall 

 not consider it here. 



