402 Journal of the Asiatic Society of Bengal. [August, 1908. | 
Theorem XV.—The difference s—1 between the ——- of 
are and chord of an infinitesimal non-cyclic are PQ is 3Ra®, ne 
lecting infinitesimal of fifth order, where & is the radius of 
curvature at P, anda is the angle between chord PQ and the 
tangent a 
Divide angle a into an infinite number of small parts (say 
” equal parts where n is large), by the lines PO,, PU,, POg, ete., 
where 0,, O03, O3, etc., are points ou the are PQ. 
Then $= 2 Or-1 0p in the limit when »= # 
l= 3 (P0,—PO,-1) 
Therefore, s—2= Lt & (O,-10r + PO,-1—PO,) 
=i 2 RB (a, —a,_1)a,—14% 
=}RB Lt ¥ {a8,—a3,_, —(a,—a,-1)9} 
=1 RadS—1 R Lit 3) (@r—ay-1)8 
=1Ra'5, 
Ae 3 3 
Since Lt 3) (a-—a,_1)° = Let z Ja (—) = Lt —=0. 
n 
Cor. A.—The difference s—J is independent of 8, if we neglect 
infinitesimals of fifth lore R and a being given. 
Cor, B.—Sin 6=0- © G neglecting infinitesimal of fif,, 
order. 
Cor, O.—Area of segment, bounded by s and 7, 
=2R*3{(a,—a,_1)a,a,_1 + 2a,a,-1(a?,,—a%,_1)tand} (by Theorem XII) 
= 2R? {403+ a* tand} 
For, (ap —a’—!)a,a,_) = 408 
and %2(a%, —a®,_1)a,a,_;= S{at, —a+,_1 — (ap—a,~1)9(a,+ ap-1)}=a* 
N.B.—If only the radius of curvature be finite and con- 
tinuous and not also its partial rate of variation, then it is more 
easily shewn, by omitting tan 8, that s—J is equa al to 4 Ra’, where 
we neglect infinitesimals of the toro order not fifth. The writer 
is not aware of these rigorous geometrical determinations having 
i ge 
ade re. 
ally by stating that the difference is of the third order, 
Oy 

