DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 393 
measured from the vertex, a series of lines in geometrical progression, 
g (sec0+tan0), g (sec 04 - tan0)^, g (sec0+ tan0)® g (sec0-|-tan0)”. 
Join N, the general representative of the extremities of these right lines, with the 
focus O. Erect the perpendicular OQ, and make the angle NOT always equal to 
the angle NOQ. The line OT will be =gsec0, the line OT^=gsec (0-^6), the line 
OT,^=gsec &c., and we sliall likewise have 
AT=^ tan0, AT^=:^ tan ATii=g tan (9-^9-^0), &c. 
This follows imm.ediately from (342.); for any integral power of (sec0+ tan9) may 
be exhibited as a linear function of sec 0+ tan 0, if © = 9-^9-^9 . . . &c., 
Fig. 24. 
P, 
since sec {9-^9-^9-^9 &c. to w0) + tan (0-*-0-1-0-L0 See. to w0) = (sec 0+ tan0)". 
Hence the parabola enables us to give a graphical construction for the angle 
(0-i-0j_&c.) as the circle does for the angle n9. 
The analogous theorem in the circle may 
be developed as follows : — In the circle OBA, 
(fig.24) take the arcsAB = BB^=B^B^^=B^jB,;; 
...&c.=2&. Let the diameter be G. Then 
OB = G cos^,OB^=G cos2^,OB^^=Gcos3^... 
&c. and AB=Gsin&, AB^=G sin2S^,AB;^=G 
sin3^ ... &c. 
Now as the lines in the second group are 
always at right angles to those in the first, 
and as such a change is denoted by the symbol 
v/— l,weget OB+BA=G (cos^+x/^ sin^), 
0B^4-E/A=:G (cos2S'4-\/ —1 sin2S^) = G (cosS-4-\/ —1 sin^)^; 
OB,^+B^,A=G(cos3S^+\/ —I sin3^) = G(cosS^+\/ —1 sin3-)^&c. 
LXV. The known theorem, that a parabola is the reciprocal polar of a circle, 
whose circumference passes through the focus, suggests a transformation, which will 
exhibit a much closer analogy between the formulae for the rectification of the para- 
bola and the circle, than when the centre of the latter curve is taken as the origin. 
Let OBA be a semicircle, let the origin be placed at O, let the angle AOB=^,and 
let G, as before, be the diameter of the circle. Through B draw the tangent BP; let 
fall on this tangent the perpendicular OP=jo, and let BP the protangent be equal to t. 
Now as p=G cos^^, and ^=:G sinJ^ cos^, as also the angle AOP=2S^, if we apply to 
the circle the formula for rectification in (33.), we shall have the arc 
AB=5=:2(^cos^^d^— G sin^ cos^. 
Make the imaginary transformations cos^=: sec0, and sin^=^^^tan0, and we 
shall have 
The expression for an arc of a parabola, diminished by its protangent. 
3 E 2 
