658 Prof. H. H. Jeffcott on the Whirling Speeds of a 



Substituting for Y and Y' and reducing, we find, from 

 OtoP, 



12E17 2 /' {fil TV) u = 2U'(/jlI+ l')z i& 1 (l-z 1 ){z£2l--z 1 ) -z 2 } 



+ M 1 {/ 2 -c 2 -3(/-, 1 ) 2 }] 

 -^(/-^)(2Z-, j [X 1 , 1 (/-.e 1 )(9/_c l) -M 1 {3(/-c 1 ) 2 -^}] 

 ^l z (l- z )(2l-z)^z^l'^z^{2l'-z^-M^{^(l'^z^-n]; 

 and from P to L, 

 12EIW( / aZ + 0» = 2«'(^+/)(/-;)[X 1 - 1 {-(2/-.-)-- 1 2 } 



+ M 1 {3..- 1 2 -.-(2/-c)}] 



- M l'z(l-z)(2l-z)[X 1 z 1 (l-z 1 )(2l-z 1 )-M 1 {i(l-~i?-P}] 



+ fc (/_,)(2^-,)[x 1 V(r-, I ')(2r-, 1 ')-M 1 '{3(r-c 1 ') 2 -r 2 }]. 



From to F, 



+M 1 '{r--' 2 -3(/'-~- 1 ') 2 }] 

 -//(r--')(2r-•')[x 1 '- 1 '(r-r 1 ')(2/'--/)-M ] '|3(^ / -- 1 r-/' ;, }] 



+ ll ,lz'(l'-z){2V-z ! )^ 1 z l (l-z l ){2l-z l )-M l ^{l-z l Y-P\]. f 

 and from P' to L', 

 12El/ J .ir-(/ J ,l + l')v = 2U'( /J .l + r)(r -:')[X 1 ': 1 '{z'(2l' -:') -Z! 12 } 



+ M/{3x I '--.'(2/'-:')}] 

 -lz'(l'-,')(2l'-z')[X 1 W(l'-- 1 ')(2r-z 1 ')-M 1 'mi'-^-P\-] 



+ ^z'(l'-z'X2l'-z')[X y:i (l-z 1 )(2l-z 1 )-M l {3(t-^y-P}^ 



If, now, we take a number of loads, and put X 1 = 7» ] &> 2 i/ 1 , 

 X 2 = ??i 2 ft) 2 z, 2 , X/ = ?7i 1 '©H' 1 , etc. ; 



M 1 = mAv( g^, M 1 ' = m/AV 2 o, 2 (g) i etc. ; 



and sum up for all the loads, we find, on putting l' = \l, 

 Z = tl, z' = t'U ; 



£P r 



■^pv r =(l-t,)-2'j[mutt{t r (2-tr)-t*} 



+ m P*{3( ! -(,(2-*,)}] 



+au»™/(i-o{'(2-o-^} 



+ mP < l l "{l-t;-'i{l-ty\} 



