of Edinburgh, Session 1882 - 83 . 
175 
but 
wherefore 
dy = cos a . dl 
X. dy = c^.coB a. da', 
but the product x . dy represents the increment of the area BOHP, 
wherefore we have 
S = c^.sina . . . (3) 
whence also 
S = . sin A . 
When the angle A is acute, as in fig. 7, there is no difficulty in 
interpreting this equation j but when that angle is obtuse, as in 
fig. 8, we have to observe that, in supposing the point P to move 
along the curve from B, carrying with it the ordinate PH, the area 
OBPH increases until P arrive at D, where the curve becomes per- 
pendicular to the axis. Beyond that limit the motion is towards 
BO, the differential of the area becomes subtractive, and thus, for 
the point p in the figure, s would represent the surface BPD^HO, 
while S would stand for the area BPD^^AO. And when the spring 
is crossed so as to take A to the other side of 0, as in fig. 9, the 
surface AFO must be regarded as subtractive, so that the symbol S 
then stands for the excess of BDF above FOA. 
On inserting the value of dl, viz.. 
- dl = dx. sin a~^ 
into the equation (2), we obtain 
- xdx — d^. sin a. da 
which, when integrated, becomes 
— -}- c^. cos a -f constant . 
How, when a? = 0, <x becomes A, wherefore the value of the con- 
stant is - cos A ; so that 
= c^(cos a - cos A) , . . (4) 
which equation completely determines the relation of a the inclina- 
tion, to X the ordinate of the curve. 
Here we observe that when P is at B, a becomes zero and x 
becomes X ; wherefore 
JX2 = c\l- cos A) , which gives 
X2 X2 
.2 — 
2(1 - cos A) 4(su) JA)2 
( 5 ) 
VOL. XII. 
M 
