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arm AC, and in a contrary direction. The number of teeth on HEO 
is twice the number on NLF. Hence NLF makes two revolutions 
round its axis, while the arm causes it to revolve round the fixed 
centre-circle HEO. The axis C is rigidly connected with a wheel 
XWP, revolving on the upper surface of the plate-arm ABC, whose 
circumference continually passes through A. Thus while the plate- 
arm ABC makes one revolution round A in the direction of the 
hands of a watch, XWPA, which is rigidly connected to the axis 
C of NLP and partakes of its motion, makes two revolutions in 
the opposite direction. Thus it revolves round an imaginary 
annular wheel XYZQ, whose diameter ZQ is double that of its 
own ; and in virtue of hypocycloidal motion, if XWP be the 
initial position, the point A in the circumference of XWPA will 
move along ZQ. 
In the annexed figure (fig. 9), the modus operandi of the machine 
is sketched. The system is analogous to the sun and planet wheel 
invented by Watt. 
It would be going beyond the limits of this paper to direct 
attention to the two modifications of the Peaucellier cell, which 
have been called respectively the Quadratic-binomial Extractor, 
and the Conicograph. The introduction of these into the sphere 
of mathematical investigation has given several indications of 
valuable results to be obtained therefrom. For a brief outline of 
these, we refer the reader to the pamphlet of Professor Sylvester, 
who not only was the first in this country to direct attention to 
the general problem, but also had the credit of demonstrating the 
