
400 » Journal of the Asiatic Society of Bengal. Dahan a. 
Cor. B.—The complete rate of variation of the radius of 
curvature at any point P, of a convex are, is ey 5, where 8 is 
the angle of aberrancy at P (Transon’s 'Theore 
For es pies ark variation of , circle ee curvature PPP 
Theorem X.—If PT, 1'Q be etce. at Pand Q to a posi- 
tive non-cyclic infinitesimal are PQ, the difference of P7’ and TQ 
is ultimately equal to 2Ra® tan 8, where 8 is the angle of abér- 
rancy and R& the radius of curvature, at P, and a the angie 
1 
For, if B be the angle PQT, then 
PQ 
i ie sin B_2sina _ radius of circle PPQ 
TQ... sma PQ radius of circle PQQ’ 

‘Therefore, nas 
TQ-P! radius of PQQ—radins of circle PPQ 
TQ+PT radius of PWY+ radius of circle PPQ 
TQ-PT = as é 
PQ 
or, scot TQ—PT =2 Ro® tan 3. 
Cor. A; eng aig ieee 2a tan 6, - 



, ultimately 
Theorem XI.—If O,, QO, Og be any: thibe points- on the 
positive non-cyclic infinitesimal are POQ, then the radius of the 
_— O0,0,0; is equal to ShO. foe tan 6}, where 
: i PO,, PO, make with the 
jenpent at P, 6 the angle of aberrancy and R the radius of cur- 
vature at P. 
For, the radius of circle 0,0,0, is evidently 
R+(PO,+P0,+PO;) tand=R+2R (a,+a,+a,) tan 8 
PO, _ PO, = 298; 
a a5 
Theorem XII.—Ié 5 and / - the lengths of the are and 
chord of any positive non-cyclic infinitesimal are PQ, then s=1 
=2R (a+2a? tan 8), where 8 is the angle of aberrancy and & the 
radius of curvature at P, and a the augle which the tangent at 
P makes with PQ. 
since 2h — in the limit. : 

4 Th 1e above simple and general demonstrati on of Transon’s ggg ee is 
14 
based on the conception of partial rate of variation of curvatar ranson 
himself deduced his theore p rties of conics ( Yaceeiin Vol. vi) 
Some elegant demonstrations h by Dr t pa- 
2% aya in his —— On the nh fer eg rate of all Parabolas. (Journal 
