58 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
a common hyperbola, of a spherical ellipse, of a spherical parabola and of a circular 
arc, multiplied by constant coefficients. 
LXXXVI. There is one particular case when the area of the logarithmic hyperbola 
may be represented by a very simple expression. Let b=b, then if we turn to (448.) 
V = a 2 -f- If, and 1=1, since i— 0. Hence (452.) may be changed into 
3(AT) = a a 2 - \-b 2 tan 9-\-b 2 tan -1 ^^y== tan0^ 
+ h ' tan " („✓*»+ y sin(! cos0 ) - v tan "‘ G si n0 ) ; 
and this expression may be reduced to 
3(AT)=a V d 2 -\-b 2 tanO-\-b 2 tan -1 ^ tan O^j—b 2 tan" 1 ^ sin9^, .... (455.) 
a value entirely independent of elliptic integrals, and which may be represented by a 
right line and the difference of two circular arcs. 
LXXXVII. The curve of symmetrical intersection of a sphere by a paraboloid, 
whose principal sections are unequal, may be rectified by an elliptic integral of the 
third order and circular form. 
2 2 
Let x 2 -\-y 2 -\-z 2 =2rz, and y-\-'j~ = 2z (456.) 
be the equations of the sphere and paraboloid. Then finding the values of dx, dy 
and dz, 
( ds Y__ {r^—kk x )z—2r{r-k)(r-k x ) 
\dzj z[z — 2{r—k)\[2{r — k x )—z] \ * ') 
Assume z=2(r— k) cos 2 0 + 2(r— b x ) sin 2 0 (458.) 
Introducing the new variable 6 and its functions, 
ds ^ ^k x (r— &) 2 + £(r— £j) 2 tan 2 # 
V(r — k) + (r — k x ) tan 2 # 
Assume b(r—b x ) 2 tan 2 9=b 1 (i — b) 2 tan 2 p, 
then introducing the variable <p and its functions, 
ds Vkk x V (r— k){r— k-^) 
dd */k(r— /q) cos 2 <p -f- k x [r— k) sin 2 ip 
and 
Hence integrating, 
V 
dk x {r — k) 
k-L 
sin 2 f 
d0 m 
dp' 
r—k x 
d k x (i — k) 
V k(r—k-d 
(k—kj (r 2 — kk x ) . 2 
1 — - — A -T-A'sin^ 
k (r— k x y 
k x (r — k)§ r dip 
*/k(r— A X )J [l— msin 2 <p] d\ — « 2 sin 2 <p 
(459.) 
(460.) 
(461.) 
(462.) 
