324 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
(f-) 
or as 
dx dx J 
Cg.) 
Adding this equation to (e.), we get for the final result, 
(33.) 
Throughout these pages, to avoid circumlocution and needless repetitions, we shall 
designate as the j»ro-jected tangent, or briefly as the protangent, that portion of a tan- 
gent to a curve, whether it be a right line, a circle, or a parabola, between its point 
of contact, and a perpendicular from a fixed point let fall upon it, whether this per- 
pendicular be a right line, or a circular, or a parabolic arc. This definition is the 
more necessary, as the protangent will continually occur in the following investiga- 
tions. The term is not inappropriate, as the pro-tangent is the projection of the radius 
vector on the tangent. 
XIII. To apply the formula (33.) to the rectification of the spherical ellipse. 
Let, as before, A and B be the seraiaxes of the plane elliptic base of the cone, r the 
central radius vector drawn to the point of contact of the tangent t, p the perpen- 
dicular from the centre on this tangent, t the intercept of the tangent to the plane 
ellipse between the point of contact and the foot of the perpendicular, X the angle 
between p and A. Let a, |8, p, r be the angles subtended at the centre of the 
sphere, whose radius is 1, by the lines A, B, r, p, t, we shall consequently have 
A=A:tana, B=/ctan^, r=^tanp, p=^tanra', and ^=\/A’^+pHanr. . . (34.) 
Now in the plane ellipse 
p^—Ps?‘ cos®A+B^ sin% 
therefore in the spherical ellipse 
whence 
tan^B5“ tan^a cos^X-f- tan^jS sin^X; 
secV=sec^a cos^Ti-h sec^(3 sin^X. 
(35.) 
Dividing the former by the latter, 
tnTi^« ons^X 4- sin^X 
Introducing this value of sinks' into (32.), the general form for spherical rectification, 
the resulting equation will become 
XIV. To reduce this expression to the usual form of an elliptic integral. 
Assume 
tan%=cos£tanX 
(38.) 
