

Vol. 1V, No. 8.] Geometrical Theory of a Plane Non-Cyclic Arc. 399 
[N.S] 
The angle between the normal and deviation axis at P, both 
drawn outwards, is called the angle of aberrancy at 
Theorem VII.—1n any convex infinitesimal aro. POQ, the 
supplement 6, 5 —- angle POQ, and the 1 
the tangents at P, Q make with PQ, are infinitesimals of the 
first order and ress equa 
For, if RB, BR, Rg be the radii of the circles POQ, PPQ, 
PQQ respectively, ‘then hk, &,, R, are finite and ultimately equal 
to the radius of curvature at P. 
But, PQ=2H sin 6= OR, sina=2h, sinf. Therefore, 0, a, B 
are San sana infinitesimals of the first order 
r, A.—If Pi?’ and QT be tangents at P and Q, then PT and 
QT are ultimately equal, and the radius r of the circle PQT is 
ultimately equal to half the radius of the circle of curvature 
a 
or. B.—The difference between the are PQ and chord PQ 
is less than a quantity which is an infinitesimal of the third order, 
For, the convex arc +o, falling inside the triangle PTQ, has 
length between PT +TQ and PQ. Hence the difference between 
the are and chord is less than PT+7TQ-—PQ or 8r sin 3 
B 
sin sin 
ry 

> = which is again less than raB (a+f). 
—The difference between 6 and sin @ is less than a 
poe! orca is an infinitesimal of the third order, 6 being of 
e first or 
Panes ‘VIL. .—The angle of aberrancy, at a cyclic point on 
@ convex ae, vanishes, 
Le be a cyclic, point. tre any pia arc POQ. 
Then, from Cor Theorem IV, a smaller P’OQ’ can be 
always found, ee "that the eis P’T and "@ T' at P’ and = 
are equal, Therefore if R be the middle point of P’Q’, T 
is at right angles to P’Q’. Now, 7’R becomes the deviation axis 
at O, ultimately. T “(ln Wi the deviation axis a 0 —— with 
the normal at QO, and the angle of aberrancy v 
Theorem IX.—The partial rate of vninitiie of. tie radius of 
curvature, at any point P of a non-cyclic arc, is tan 3, where 3 is 
the angle of aberrancy at 
Take an infinitesimal : are PRSQ, where RS is par allel to 
PQ. Then, from Theorem VI, we have tan b= 5) where UV 
is the distance between the centres of the circles RSQ and PRS. 
ow, it is easily seen that UV is ultimately equal to the differ- 
ence of the radii of the circles RSQ and PRS. Hence, tan 8 is 
io to the partial rate of variation of the radius of curvature 
Cor, A—If PQ be an infinitesimal = arc, then the 
difference between the radii of the circles PQQ and PPQ is PQ 
tan 8, for the circle SHS is transformed into the circle PQQ by a 
single change of P into Q. 
