ATTRACTION IN THE CASE OF THE ENGLISH ARC. 
41 
Now draw FGHG' parallel to the axis of?/, and let AH=^. Then the attractions of 
tabular masses on ¥b and Fb'— those of the masses on Fb and Fb'— those of the 
masses on EG and EG' : hence 
Attraction on A parallel to x of the tabular masses on Fb and Fb'— 
log. 
and fli log. 
+ 1 
These may be written in the following form : — 
f H log. 
V‘*S*5 \/'*S+l 
and fH log. 
If the numerator and denominator of the fraction under the logarithm in the second 
of these be multiplied by 
its value will not be changed, but it will become 
fH log. 
Now this is precisely the same as the first when —y is put for y ; and it is only in 
this change of sign my that the parallelogram F^' differs from Fb. Hence the first 
formula includes the second, and is applicable to all tabular masses lying on the right 
(or the positive side) of the axis of j/, on either side of x. It will be observed, how- 
ever, that the same is not true of Y in this formula, as may easily be seen by going 
through a process similar to this transformation for ?/. We must therefore remember 
not to let Y be negative. The way to obviate this in the use of the formula is to 
make the direction in which Y is measured in any particular case the direction of 
positive ordinates for that case, and then y will be positive or negative as it lies on 
the same or the opposite side of the axis of x from Y. 
16. The formula can be very much simplified as follows. 
Put — = tan ^ 1 , ^ = tan 0^, ^ = tan 6.^, - = tan 6 ^ ; 
Y 1 + sin 5, 
X cos Sj 
— tan ^45°-j-2 
and similarly of the others. 
MDCCCLVI. 
G 
