DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 353 
This is a very important theorem, as the protangents are arcs of equal parabolas, 
all measured from the vertices of the parabolas. Hence also the length of the protan- 
gent arc depends solely on its normal angle. 
As an arc of a circle may be expressed by the notation 5=sin“*^|], y being the ordi- 
nate and A- the radius, so in like manner an arc of a parabola may be designated by 
the form ^=tan“*0j; y being the ordinate and A the semiparameter. To distinguish 
the parabolic arc from the circular arc, the former may be written s=7av~'^(^. Again, 
as we say, in the case of the circle, the angle u and the arc ha, a being the angle con- 
tained between the normals to the curve at the extremities of the arc : so in the para- 
bola, we may write a for the angle between the normals, and (h.a) for the corre- 
sponding parabolic arc. In the case of the parabola the arc is always supposed to 
be measured from the vertex ; in the circle the arc may be measured from any point, 
as every point is a vertex. 
XL. Resuming the equation (157.)5 A2=JdAv/P'-}-Q'sin^A-l-R'sin^X— We 
shall now proceed to develope the first integral of the second side of this equation. 
As the integral is precisely the same in form as (123.), and the amplitude as 
also the modulus i{=i, we may substitute a, jS, y for A, B, C, m for n, for 0„, 
retaining the modulus and amplitude, which continue unchanged, as we have esta- 
blished in (152.) and (154.) ; or substituting for a, jS, y their values in 7n and i, we get 
2 — 2m] 2 
= —m^„ 
[i^ + m^ — 2m] 
m 
If . 
J [1 — m SI 
df 
sin^f] \/l— ^■2 sin^a 
-m]f c]<p . C, /-. [f + m^—2m]2 f dr 
?■ • 
(161.) 
If we eliminate i from the coefficients of (133.) and (161.), putting M for (1 — msin^ip), 
and N for (1— wsin^^), as also for 1 — ? sin^ip ; (133.) will become 
and (161.) will be transformed into 
2(n — m) 2 ^ (1— m)(w— m)f d<p , n /t 2(n — m)(^ dr 
^ = (163.) 
If we compare together (162.) and (163.), which are expressions for the same arc of 
the logarithmic ellipse, and make the obvious reductions, putting for and their 
values and shall get the following resulting equation of 
comparison, 
/l— wxT d<p ^ 2 r dr 
\ n m /jMi/I—mnJ \/l~ -v/^Jcos^r 
sinip cos<p I 
MN 
(164.) 
2 z 2 
