DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 401 
If, in these equations, we change n into —n, and therefore sin| intOAy/— 1 tan©, 
sin^ into \/— 1 tan©', 
sinr into \/— 1 tanr, sinr^ into \/ —1 tanr^, and sinr^^ intO\/ —1 tanr,^ 
the preceding equations will become 
-v/mn sin<p sin;,' sinw tan©' ^ 
COS<p COSX, COSW 
l+I 
tanr= 
1+71 
mn sin;p cos^ \'^mn sin;)(^ cos^;' 
Vl — 7^ sin®(p ’ ’ 
cosip cos;)^ cosoj 
(383.) 
tanr„= 
a / 7nn 
sinw COSO) 
J 
and © + 0 =r4-r^ — as in (360.), values which coincide with those found in LXIX. 
for the circular form. Or we may pass from the logarithmic to the circular form, or 
from the paraboloid to the sphere, or inversely, by the imaginary transformations above 
referred to. 
We shall find on trial, that the angles v, v and r in (279.) fulfil the condition of 
conjugate amplitudes. 
Section IX . — On tlte Maximum Protangent Arcs of Hijperconic Sections. 
LXXII. Since the protangents vanish at the summits of these curves, there must 
be some intermediate position at which they attain their maximum. When the curve 
has but one summit, as is the case in the parabola, the hyperbola, the logarithmic 
parabola, and the logarithmic hyperbola, there evidently can be no maximum*. 
In the plane ellipse, the protangent t-- 
ai^ sin<p cos<p 
v/1- 
SlQ'^ip 
If we differentiate this expres- 
cb 
sion with respect to <p, and make the differential coefficient ^ = 0; we shall get 
tan(p= — ^ 
^3 
Substituting this value of tan^ in the preceding expression, 
t—a — b 
(384.) 
(385.) 
In this case, the arcs dravvn from the vertices of the curve, and which are compared 
together, have a common extremity, or they together constitute the quadrant. 
The coordinates t, y of the arc measured from the vertex of the minor axis 
'f! h 
are .r=a sin^, 7/ = 5 cos^, therefore cot^=j cot^, since ja=b. If we now make 
cot^=vv 5 x~j^’ tanX=p tan?i; or making X=S‘, or tanX=:-^. 
* The investigation of these particular values of those portions of the tangent arcs to the curves, which lie 
between the points of contact and the perpendicular arcs from the origin upon them — or as they have been 
termed in this paper, protangent arcs — is of importance ; because, as we shall show in the next section, in the 
different series of derived hyperconic sections, the maximum protangent arc of any curve in the series, becomes 
a parameter in the integral of the curve immediately succeeding, 
3 F 2 
