195. .'rth\VtK ^VJ0<iK5ii\1^ftli] iiWXi ^tiK^^s^^-iVwl 



,M?,v,I,f ..^ :(: jtyyir;. ?.h iTheorem^. , *,,,! ........... . -;r;-.;a 



pORTION!8iio£ithe'«bove paper (vide Phil. Mag,, p. 286^ 

 -■- for October last) possibly requiring iurther elucidation, 

 it may be observed that the conclusion in the odd case is better 

 proved as follows:— ^nuvjjj^yw 



As the binomial theorem for whole powers e x tends 'to? frad-- 



(12 4^1 \ 

 e. g. — (- - = ] , — \ = 7 ) j 

 3 3 5 5 / ' 



the equations between 'fe, q^,, Ti, Y, exist if even these quanti- 

 ties are fractional. Let k be any whole number, and make 



.nz'.ria 

 multiplying by 2 ^-^ikm-m ^nd then by a'^'*'^¥, - oJ BS sni J 



.ama-uxa laiiiia mMi iuo -gahi^z ^a^tcinq^s Ifiimonid stli 

 \6 tiaiil 9rfj»moii aldiywb^h^YlaJEibamTgps? ~D?*:H<n{T .8.^ 



.19 w&f a jw S^ »to+<s T^^rfilbi Ijj^-^I • -J J. msbhiio-y 

 -oM$<i -io Mmiaj^:i on pi fjayjo^pm giatjiuj a^Mim/'i fiv:Li 

 = \p 2r*+ Y + gry +-2 / + \;? ^*+ 1 — qj/^:A^A\ 

 .a .M .B ^ , ^ , \» / 1 1 \? 



The foot indexes of B ought to have been doubled, and the 

 equation transforms to •\^.k\ a /I 



f)?o ri J • gij^ *8i&^^ ( iiw J ^' + 2 , _L , ^ -Sraf figroc^j/^i *A bBil rfniiiv/ 



iioiiJSJigB ^ 8«M-lij4»- «*^»-— o — ( ) "^ -^'T^' 4**' ""^•'^ "^''''l 



jAf being independent of ^, w = 1 gives p = l z=qi n -- i 

 enters all B,- except Bq = 1, .-. = 1 — (^ ^yf is the equation 

 in this case. If k = 0, z and j/ are indeterminate ; tor any 

 other value of ^^ ^=j/, and therefore jr = 0, an excepted case. 

 When w > 1, and ? > 0, all the terms are negative, whatever 



^■»ri.1 ui 89vi'n/5 gijd1 hrr^ .h'ld'insdn ylrfo ?i -t:^ " •■■.■■ odi xi'Jiv/ /Ur/j 

 'duH niahSD lo noiff; Commtmicated by the Authon. rr ^hz89r Tialliqao 



