i 
— &@=a+be+cx*, theny= /u=u and dy = ful? = Junt= = 
; du 
and radius of the circle of curvature are called 
PRELIMINARY 9 
1 
i Jatteteat’ 
Also, SY = 0 + 2ex, therefore ay __ b+ dex 
; dz 
| 2 Jat ba + cx” 
Successive differentiation.—If y is a function of «, and o =u, where 
Fig 
u is also a function of x, then mo, where v is another function of «. 
du dt. 2 
Hence, 7 dx, and this is written et 
—— ¢ la dx 
The integral of the sum of a. number of functions is equal to the 
sum of the integrals of the functions, Thus, 
(ax + ba")de = [ae + [eae +C, 
where C is the constant of integration. 
9. Circle of Curvature.—APB (Fig. 2) is any curve. M and N are 
two points on this curve, on opposite sides of the 
point P. A circle may be drawn through the three 
points M, P, and N. If the points M and N be 
moved nearer to P, then when M and N are inde- 
finitely near to P, the circle becomes the circle of 
curvature of the curve APB at P. The centre 
the centre of pitoatene and radius of curvature 
respectively. or “ 
Let CPD be the eisile of curvature of the Fic. 2. 
curve APB at P. Let X and Y be the co-ordinates 
of the point P, considered as a i point on the circle CPD. Then, 
(X-a)?+(Y¥ —-b?=R* 
Differentiating once, (X —a)+(Y—- wr : aohe 
2 
Differentiating again, 1 + (%) + (x- - “ex otk: 
Let x and y be the co-ordinates of the point P, considered as a point 
on the curve APB. Then X=, and Y=y. Also, since the circle CPD 
and the curve APB have the same tangent at P, ay and “Y denote 
dX dz 
the slope of this tangent, therefore e a. Lastly, since the circle 
: c 
CPD and the curve APB have the same curvature at P, and since 
curvature is measured by the rate of change of the slope of — 
ey _@y, ly (2 y pty —0. 
aX2 de? hence, z-a+(y—- t= 0, and 1+ +(y—b)— 78 
: 
1+(# * dy { + (92) ; 
det da\ ~~ \dx} J, 
: Therefore, y-b= == Oy and 2-—-a=-—— By ee. 
