DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 327 
fall the perpendicular arc Zt on the tangent arc of the circle, making the angle X 
with the arc Za. From O let fall on the tangent to the plane ellipse at Q, the 
perpendicular OP making the angle with OA. 
tan^a , , sin^a , 
Ihen tanX=^^ — ^ tany, and tanX,=^-^ tand/. 
Hence we derive Whence tanX.tan>L,= cos^£tan^>i. 
tan A ' 
But we have shown in (39.) that 
tanV= cos^a tan^A, 
whence tan®(p= tan?^ tan\, (48.) 
on the tangent of the amplitude (p is a mean proportional between the tangents of the 
normal angles which a point of contact k on the spherical ellipse and its projection Q 
on the plane ellipse the base of the cylinder produce. 
XVII. We may obtain, under another form, the rectification of the spherical 
ellipse. 
Assume the equations of the right cylinder and generating sphere as given in (19.), 
Make 
hence 
and therefore 
Now 
2 2 
J+f2=l, and 
j:=asin0, y=bco^Q\ (49.) 
z^=ld—a^ sin^0— 6^ cos^0 ; 
^■dJ“L (F- 62 )cos 26 >+(F-a 2 )sin 26 > J 
a^(yP— 6^) = A:^ sin^«cos®(3, &^(F— a^) sin^/3 cos^a, P — b^=lf cos^(B, ld—a^=ldcos^cc. 
Substituting these values in (50.), and integrating. 
, r . rtan^a cos^d + tan^/3 sin^0q i 
Lsec'a cos^0 + sec^/3 sin^^J 
If we now compare this formula with (37.) and make 0=X, we shall have 
s’ — ff=r. 
(52.) 
Hence we may represent the difference between two arcs of a spherical ellipse, mea- 
sured from the vertices of the major and minor arcs of the curve, by the arc r of a 
great circle which touches the curve. 
XVIH. We may thus, by the help of the foregoing theorems, show that when any 
elliptic integral of the third order and circular form is given, whether the parameter 
be positive or negative, we may always obtain the elements of the spherical ellipse, 
of whose arc the given function is the representative. 
Let the parameter be negative. 
As 
„ tan^« — tan^/3 
and sin^;j= 
sin^a— sin^jS 
-=^ 
2 u 
MDCCCLII. 
sin^sc 
