AND VIBRATION OF SHAFTS. 
357 
the elimination of the eleven ratios 
A:B: C :D : A' ; B':C': D' : A" : B" :C" :D" 
leads to 
rdc/ 
«+ + ^. f + ft f + ' 5 >ft “f + “ 1 “^ 
+ 1 1 “ “ 1 P 2 "YT 
/, /» '3 /> /> ''S /. '2 
^ 1^1 . '' 2^2 • ^ 
Co 
I + «i + «3 + /3] - 7 - + 9 + «i 
ao 
6 ' 6 ' /-J 2 ' 2 ' 2 ■ 36 
+ “ 1/^3 ^ 
- j “iCiCi' + + ^2 + + /^i^s ^ + “1^3 
O ‘ 
^ 3„ '3 o 3^ '3 ^ 2^ •'3 » 2« '3 ^ 3^ '3/'3 ^ 2/. '27' 
■ *^1 ^1 I I I /t ^3 ^-2 1 ^2 '' I o /o ^2 '' 
6- + ^ "sT + ft ^ + ft -T ' + “1“^ - 2iF + ^'ft 
I p CicJH “ ^ t’dcj I 
in which 
24 
O’l + Cj — <^2 + C 3 — I, 
/ '7' 
C 2 C]^ - C^ C 3 = 2 = i , 
[A], 
If the second j^^dley be supposed removed, that is, if we put W 2 and I 3 each equal 
to zero in equation [A], we get 
Oi 
Wdi 
%(jWl 
2^1 
“b oi 
3 J 7 , ' 
+ i'l(3o.c/ 
0 , 
a result, of course, identical with that already obtained (Case X, § 26, p. 308). 
It will be seen at once that the equation [A] is practically useless unless some 
special relation be assumed between the dimensions of the pulleys, &c., and even then 
it would be impossible to compile a table which could be used except in very few 
cases. 
Cases, other than the above, in which a shaft is supported in a certain manner and 
carries two pulleys (for example, a span with an overhanging portion on one side 
and supporting one pulley between the bearings and another at the end of the over¬ 
hanging portion) have been investigated, and in each case the result obtained was too 
complicated to admit of any practical assumption. 
