ATTRACTION IN THE CASE OF THE ENGLISH ARC. 
39 
171 171 
I shall make m so small that the squares and higher powers of ^ and y be 
neglected. Hence these formulse become 
I 
I 
dx 
2n+ i 
Y2)2 
^ dx 
V ^ 0 A 
dx 
2 4 
2n ~ 2ra Y nd" 
and 
L=log, 
X 
m 
1 + 
^/ 
, X2 
1 + ya 
These formulse fail when n—0 ; but in that case 
dx L . 1 / 1 
I 
V3“h 
. x{x^+Y^f Y^^Y^V VX^ + Y^ Vm^ + Y^ 
2 + Y2) 
by direct integration. 
L-l 
Y3 
also 
f 
xdx 
Y2 vx^ + y^ 
1 I 
{x^ + Y'^f Vm^ + Y^ VX:^ + Y^~Y ^/X2+Y2 
Substituting these in the expression for the attraction, 
Attraction of the whole mass on A parallel to x 
h 
Y^ Y ' VX^ + Y^ 
1 h h^' 
IK- f-lLj 1 A 
+ ^ 3(^2 Y3^“2 Y mV 
+ .... 
IK I / ^ 
“•"^2m+i^2 4”' 2n 2n Y m^V 
+ 
} 
=fH[L(l+|K.^+...+|| 
3 5 2»+l^- 
Jv, 
2n 
^2»+ 
1Y2»“ J 
/I ^ 1 ^ ^ K A!_L M 
r(~ V ^^2«+i,„2n+‘-y | 5 
or, neglecting the squares and higher powers of y’ 
=fH{L-(iK,^,-iK.5+... + (-l)-iK,„,^+..)}. 
It has been laid down that h and m are both small quantities. This expression 
shows what the limiting value of their ratio must be, that this formula may be 
capable of use. 
