THE ROYAL ARTILLERY INSTITUTION. 
163 
ON THE CURVED RACK 
MONCRIEFFS PROTECTED BARBETTE GUN CARRIAGE, 
[COMMUNICATED BY JAMES WHITE, M.A.] 
The curve is traced by a fixed point in a wall upon a circular disc rolling 
against it along a right line. As applied in the carriage the rack of its 
form fastened to the elevators as they roll back turns a pinion on the rail 
by which the friction and pawle wheels are moved. But it is manifest that 
the curve is not limited by the circumference of the wheel or circular disc 
generating it; if the plane of the rolling circle (on which the curve is traced) 
be indefinitely extended, the curve might be prolonged without limit. It 
is in this general form that it is proposed here to consider it. 
It should be remarked, to prevent misconception, that part of the curved 
rack in Captain MoncriefFs carriage is generated by that portion of the 
elevators which is not circular, but as that is very small, and also the form 
of this generating portion undetermined, it will not be taken into account. 
The curve can be obtained by reversing the supposition, i.e . by considering 
the circle fixed while the line carrying the given point on its plane rolls 
round it. This will give one-half of the curve commencing from the point 
of it nearest the centre of the circle, the other half being equal and opposite. 
Thus it may be defined as the locus of a fixed point on the perpendicular to 
the normal of the involute of the circle; or, in other words, the locus of the 
extremity of a tangent of given length (p) to the involute.* 
Taking the particular case in which the length of the tangent is equal to 
the radius of the circle (p = a) the curve is the spiral of Archimedes, whose 
equation is p = aw, for that is the pedal of the involute, or locus of foot of 
the perpendicular from the centre of the circle on the tangent to its involute, 
and the part intercepted on the tangent between that perpendicular and the 
normal is evidently equal to the radius. Erom this the general equation for 
any length ( [p ) of the tangent can be easily deduced. 
* The subject of involutes has been very extensively pursued, partly in connection with the 
curve here treated of, by Professor Sylvester, Tide Philosophical Magazine, Yol, XXXVI. p. 295, 
October, 1868. 
[VOL. VI.] 
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