CONTINUOUS CALCULATING MACHINES. 
391 
then 
I = yqft 3 -f- Cl b ^ 
— i(« 3 + 3a 2 5+3ah 2 ). 
Let now the third frame, and consequently the axis of rotation of the third sphere, 
be kept parallel to the first and perpendicular to the second, as in the former case, 
with the two sets, but let now the units be taken so that the reading of the distance 
of the pointer P 3 is 
then for the travel of the wheel A over the distance Ox, i.e., the width ot the element 
of area, both forwards and backwards, the reading of the recording wheel D of the 
third set is 
_{ct + bf b 3 
“ 3 _ 3 
=i(« 3 + 3cd6+3a& 2 ) 
= moment of inertia of element MN about axis Ox (fig. 24). 
By taking, as in the previous case, 
Vi= h 
y 2 =(a+6) 
it is found that final reading of the dial of D for the whole travel of the curve 
= f (2/a—) 3 -H 3 (2/2—2/i) 3 2 /i+ 3 ( 2/2 — 2/i )2/i 3 ) ^ 
J x 1 
=if \y%-y i) d ® 
J x x 
= moment of inertia of whole area about Ox. 
Both the foregoing results may be at once proved in a more general way, thus it is 
evident that the wheel B is turning at the rate \d A where is some constant, and 
A=area of curve, 
then by means of the addition of the second sphere and roller the product is given as 
with the first, and C turns at the rate 
— kxy{kidjK) 
3 E 2 
