THIS ROYAL ARTILLERY INSTITUTION. 
165 
(2) Let p — a, or cl = 0, tlien as before seen the curve is the spiral of 
Archimedes. 
(3) Let p — 2a, or cl— — a, and then the curve takes a remarkable form, 
which may be called the counter-involute. Taking its rise from B (the point 
diametrically opposed to A, see Tig. 1), its equation will differ from that of 
the involute commencing at the same point only in the sign of cos -1 —, 
but its form will not be at all similar. The involute may be defined as the 
locus of a point taken on the tangents to a circle making the length of the 
tangent equal to the length of the arc from the point of contact to a fixed 
one on the circle. The counter-involute admits of the same definition; but 
the tangent is taken in the direction from and not towards the fixed point 
on the circle. It may be described as traced by the extremity of the line 
round the circle when rolled off backwards tangentially, instead of unwound 
directly, as in the case of the involute. This will be easily seen from the figure. 
Let VS == QS, and V is a point on the curve; draw VP' a tangent, and 
PCP' will be a diameter of the circle; produce QC to meet VP' in Q' , and 
CQ' will equal CQ, and the angle BCQ! will equal ACQ ; therefore Q r is a 
point on the involute commencing at B; and it is manifest that VP’ is 
equal to QP, which is equal to Q'P*. 
The involute and counter-involute may be described as tan = s ; s being 
the arc of the circle from a fixed point to the point of contact of tangent. 
The Moncrieffian curve generally may be described as tan = fa; when 
and s is the arc of a circle whose radius is d. This is evident from 
d 
Tig. 2, 
BE = PQ= PA = LB A’-. 
Fig. 2. 
It may therefore be defined as the locus of the extremities of tangents to 
a given circle whose length is a given multiple of the length of the arc from 
their points of contact to a fixed point on the circle. The radius of that 
circle will be cl of the equation given before. When p is greater than a, 
i.e. when d is negative the tangents must be measured from and not towards 
the fixed point on the circle, and the curve will resemble the counter* 
involute. By a common property of rolling curves it follows that the 
normal of the curve is PR in Tig. 1, the line joining the point of contact of 
the normal of the involute with the generating point. It will also be 
observed that the complete involute of a circle (i.e. when unrolled on both 
