42 pitcher: interrelations of 



(d) Ho'"' . = . Mo 3 (3 Ms 9 e : => : 3 w," ' K,.^p,,„. . d . 



A(jUop, — Mop.) ^ eAAtspi). 

 By (4) there is available a ^s such that 



(e) Kp,„, .3. M3p ^ 1. 



(.J J -^IHin ■ •'^pipmi- • ^ ■ ^MOp ^ o. 



■"-piiumji" • 3 • A/Xop S S". 

 (^) W.2 ^ Wg, W?j, T?)!' . A',„„j„,, . itp„„ . 3 . Ahop ^ 2 



■"-pmimj • 3 . ^l/Jop ^ 2"- 



(^0 mi ^ ms, w;, ?«;' . K,„„^„, . d . ~Kp„„. 



{i) m ^ W2.3. i^.."/"'"'-^"'-- 



(12) 1^ . A' . gfjj'-^^'^'^'-^^' . >p" . gjj'"" ; -, ■ gBo(oj;'i'i"). _ _, _ gK^-m'im") ^ 



Proof.*— 



(a) t '^l'"'"'' ■ W"' : 3 : 0«»(^''=''"'- . 3 . e'''''"'^'^"\ 



(b) e''^''"'"'"' . =. 9 (3 mI m'i' 9 f : => : 3 m'. j X;,„,,, . 3 . 



Ae^y. ^ e.-iMi^.M;;- (P"))- 



(C) (/X; ;i;')'^='^"'' -'-'•"'' . = . mI y"|' 3 (3 M2 Mo' 3 f : =1 : 3 '"I' 5 A'pi'P:'m," ■ 3 . 



•^(mim'M'u/' — mU'MI;'-") ^ C-'iM-W M'>" f?^"))- 



-^'pi'P2-mi" • => • -4jul;,.' Mlp" ^ ^Mlpi' Mlp" + ^Alip,' M2p" (P")- 



(d)t 3 M^Mo' 3 (e : 3 : 3 //;, s A'',.„, . iC,,,.,,,,, . A','„.^,-,„_, . 3 . 



A0^,y, + ^e^,v, ^ eA/x;,,.,. Mo;." (?/')) • 



(/)ll TO.3.0^_/="^'^-™<-^"\ 



* In case the development A' of ^' is such that for every m the class ^'_,„ '^ ''^^ """ 

 class, (12) is at once seen to hold in \-iew of (2). 

 t § 29II. 



t By (b), (c) and 9Ji"^'. 

 § I. G. A., §73.12, 02. 

 II By (7) and (12e) . K'^ is relative to a'„ . 



