388 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
will enable us to express an arc of a parabola, taken anywhere along the curve, as the 
sum or difference of an apsidal arc and a right line. 
Thus let ACD be a parabola, O its focus Fig. 21 . 
and A its vertex. Let AB=(A:,(p), AC = (/r.xi), 
AD =(/<:.(«) and 
yy'y" 
■h. Then (343.) showsthatthe 
parabolic arc (AC -|-AB)= apsidal arc AD— A; 
and the parabolic arc (AD— AB)=BD=apsidal 
arc AC+Zi. When the arcs AC', AB' together 
constitute a focal arc, or an arc whose cord 
passes through the focus, and h is the 
ordinate of the conjugate arc AD. Hence we 
derive this theorem. 
Any focal arc of a parabola is equal to the dif- 
ference between the conjugate apsidal arc and its 
ordinate. 
The relation between the amplitudes (p and u in this case is sin2ip— ^ 
2 cosco 
COSco 
Thus 
when the focal cord makes an angle of 30° with the axis, we get cos<y=g, ov y=bk. 
Here tlierefore the ordinate of the conjugate arc is five times the semiparameter. 
LX. We may, in all cases, represent 
ordinates of the conjugate parabolic arcs 
Let ABC be a parabola whose focus is 
k 
O, and whose vertex is A. Let AO = o-= - ; 
moreover let AB be the arc whose am- 
plitude is <p, and AC the arc whose 
amplitude is x- points A, B, C 
draw tangents to the parabola, they 
will form a triangle circumscribing the 
parabola, whose sides represent the 
semi-ordinates of the conjugate arcs, 
AB, AC, AD. 
We know that the circle, circum- 
scribing this triangle, passes through 
the focus of the parabola. 
by a simple geometrical construction, the 
whose amplitudes are <p, % and a. 
Fig. 22. 
Now Ah—gX?m<p, Ac=o-tan)(^, tan^ sec)(, cd=g tan x sec <p ; 
hence bd-\-cd—g (tan<p secx+tanx sec?)), therefore g tana>=bd-\~cd. 
When AB, AC together constitute a focal arc, the angle bdc is a right angle. 
