164 
MINUTES OF PEOCEEDINGS OF 
Let PQ be the normal, and QS the tangent to the involute of the 
circle PAD. Then, let QS=CP~a, and S is a point on the spiral 
p=zao) 3 and CS is the radius vector. Tig. 1. 
Let QP=p and PS (= a —p) = d, and it is required to find the equation 
of locus of P. CS will be its radius vector. 
CP? = CS 2 + PS* = aW + d\ 
consequently, _ 
p 2 =a 2 to 2 —-—— = co, 
r a 
but to is the angle made by CS with CD (at right angles with CA) } there¬ 
fore in terms of the angle 0 made CP (p) with CD, 
CO = 6 - z ECS —6 - sin-1 - • 
_ P 
.. - vV — d* , . d 
therefore —-- — a — sm-i — , 
a p 
, n/p 2 -^ 
! a 
Professor Sylvester has proposed to call the rack of this form the 
“ Moncrieffian,” as it has been first used by Captain Moncrieff. The curves 
in general he has proposed to call “ convolutestheir existence has been 
occasionally slightly noticed, but their properties and application seem to 
have been entirely overlooked till Captain Moncrieff employed a rack of 
this form in his gun carriage. 
Three cases of the curve are worthy of notice :— 
(1) Let p — 0, that is let d—a, and then the curve becomes the involute 
of the circle. This may be found by making the substitution in the equation 
given above, remembering that for the Moncrieffian curve 6 is measured 
from CD, but in the equation of the involute as usually given, 
a _\//> 2 — cd 1 a 
0 = -cos-i- 
a p 
6 is measured from AC. 
