398 Journal of the Asiatic Society of Bengal. [August, 1908. 
vr. A.—The distance between the centres of the circles 
PPG ‘ead PQ’Q is P’Q’ tand. This is proved in the same way as 
the above proposition, 
Cor. B.-—-lf O be i yaa of the arc POQ, and R the 
midpoint of chord PQ, and 6 the complement of the angle be- 
tween OR and PQ, then te a between the radii of the 
circles POO and QOO is P 8. 
Cor. C.—If T be the srtaewsorio of the Sy at P and 
to an are PQ, R the midpoint “ chord PQ, and 8 the comple- 
ment of the angle between 7’R and PQ, then the tienes between 
the centres of the circles PQQ aa PPQ is PQ tan 6. 
Section IT.—InrFiniTesimat Arc. 
nitions.—If a number of continuously varying quantities 
aah simultaneously, t they a 
ofvanishing. One infintesimal is of the same order as, or of ahigher 
or lower order than, another, according as the ultimate ratio 

of order one. Any es infinitesimal y is then called of 
order n, if the ultimate rati y/2" is finite, that is, neither zero 
nor infinite. In all that tea the chord PQ, of the infinit- 
esimal arc POQ, will be considered as of the first order. 
If PRSQ be an infinitesimal arc, the ultimate ratio, of the 
difference of the radii of the circles RSQ and PRS, to the dis- 
tance PQ, will be called von oo cams rate of variation of the radius 
of the circle of curvatur 
If e an ‘tiafleeimal arc, the ultimate ratio, of the 
difference of the radii of the circles of curvature at Q and P, to 
the distance PQ, will be called the complete rate of varintion, 
or cate rate of variation, of the radius of the circle of curva- 
t 
ex arc, the only supposition we will make is that 
the cian of the circle of curvature is finite and varies continu- 
ously. 
on-cyclic arc, we will make the additional supposition, 
that the peer rate of variation of the radius of curvature is 
finite and continuous. 
PRSQ be an infinitesimal are RS, any minor chord 
parallel to PQ, and M, N the midpoints of PQ, RS, then the line 
through M, N, in its ultimate position, is called the deviation ' 
axis at P. 


1 Transon introdnced the aceon deviation axis. for which Salmon sub- 
stituted ‘ aberrancy axis.” Transon called tan 3 the rate of deviation from 
circular form, an exceedingly iabrenive expression, which ‘Salaiod ent 
down to ‘aberrancy.’ Both the ee have been retained, by the present 
writer, with a slight distinction in 
iouville, Vol. vi, and Satie 8 nag te Plane Curves, page 368, 3rd 
edition.) It may be pointed out, that the definition ud bares axis 
given here, is more general in form than inet given by Tra 
