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MINUTES OF PROCEEDINGS OF 
sides from tlie same point on the circumference) has a cusp, the counter- 
involute has none. 
A brief notice of the mechanical application of the convolute or 
Moncrieffian curved rack may be added. 
By a rack of this form a moving fulcrum can be controlled or applied; 
as however the fulcrum rolls the rack will pass through a fixed point, and 
therefore will act upon a wheel or pinion. Of the moving fulcrum, for 
which Captain Moncrieff first employed this curved rack, most valuable 
and important use has been made in his protected barbette gun carriage. 
The sudden and violent strain of the recoil of a heavy gun has not only 
been subdued but also utilized; and the principle may admit of many 
further applications to machinery, subject to sudden strains and violent 
jerks. In some cases moving fulcra might be advantageously substituted 
for pivots, so that the effect of the strain instead of being confined to one 
point would be distributed over some space. In the engines of war where 
the destructive effects of great forces acting instantaneously have often 
to be provided against, this method may be found useful in many other 
cases besides that in which it was first and so successfully employed. 
This curve being of the family of spirals has an infinity of convolutions. 
Consequently the rack to govern a moving fulcrum may be of any 
length or at any distance from it. 
To illustrate the application of a rack of this form; were it fixed to 
any circle a pinion at the height p from the ground-line would roll the 
circle along it, or if the circle was the moving power as it rolled it would 
revolve the pinion. 
The manner in which this curve has been obtained has caused the 
rack to be sometimes called cycloidal. But this is a confusion of thought, 
as the cycloid is generated by a point fixed on the rolling circle, this 
curve by a point in the wall against which the circle rolls. 
A small instrument by which this curve in its various cases may be freely 
drawn has been devised. It consists of a simple artifice by which a ruler 
carrying a movable pencil is kept always perpendicular to a thread unwound 
from a circle. 
It will be remembered that the Moncrieffian curve is the locus of the extremity 
of a fixed length taken on the tangent to the involute of a circle : now the involute 
continually approximates to its tangent, for its radius of curvature is continually 
increasing up to infinity, hence it follows that all Moncrieffians derived from the 
same circle are asymptotic to the involute, and consequently to each other. 
The length of the arc of a convolute or Moncrieffian curve, reckoned from the 
apse, when p = d as calculated by Gentleman Cadet Lloyd, of the Boyal Military 
Academy, is as follows :— 
fyCl aCC 
The portion cut off by the circle is therefore 
(a-d){ I WM + log + d) + ^ (2a) ] ■ 
C *w a & s/\f^ d) ) 
which vanishes as it should when a — d = 0, or a + d = 0. 
When d = 0, (the case of Archimedes spiral) the intercepted arc is 
(V(2) + log {1 +V(2)}] a. 
