68 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
In (163.) we found for the equation of the logarithmic ellipse measured from the 
minor axis, and multiplied by the indeterminate factor Q, 
2Q2=- 
1 — m\ 
1 
l. . 
(488.) 
If in this equation we substitute for m, n, and k their values as given in (485.), 
jjl\ # # (ft 
\ v ' mnkQ with the coefficient ^ of the expres- 
sion for the lemniscate in (480.), we shall find 
0 a*(b*-a*) . 
W “« 2 6 2 + ffl 4 -6 4 ’ 
hence the last equation, substituting this value of Q, will become 
2 « 2 (6 2 — a 2 ) -^_ L ? ab(b 2 — a 2 ) v / « 2 + 6 2 ^^ 
aW + at-b 4 ~*~ S <i%* + a 4 -b 4 J 9 
✓ I 
. a 5 b Cdf a{b 2 -a 2 ) 2 Sa 2 + b\ p 
1 '[fl ! i ! + fl 4 -i 4 ]^ + 4 i J^ I b(a 2 b 2 + a 4 -b 4 ) 5 
. . (489.) 
or the sum of an arc of a hyperbolic lemniscate and of an arc of a logarithmic ellipse 
may be expressed as a sum of integrals of the first and second orders with a circular 
arc. 
When b=a, the above expression will become 
In this case the parameter of the paraboloid becomes infinite, and therefore the 
paraboloid a plane, just as the sphere became a plane in the last case; so that we 
cannot express integrals of the third order, whether circular or logarithmic, by an 
arc of a common lemniscate. 
XCIII. Fagnani, the Italian geometer, first showed that the lemniscate of the equi- 
lateral hyperbola might be rectified by an elliptic integral of the first order whose 
modulus is lie did not however extend his researches to the investigation of the 
general problem of the rectification of the lemniscates. 
Although the lemniscates may be rectified by elliptic integrals of the third order, 
as well circular as logarithmic, yet these curves cannot be accepted as general repre- 
sentatives of integrals of the third order, because in the functions which represent 
those curves, the parameters and the moduli are connected by an invariable relation, 
as in (477-) and (481.). Thus the elliptic lemniscate, whatever be the ratio of the 
axes of the generating plane ellipse, can be represented only by a particular species 
of spherical ellipse, that whose principal arcs are supplemental. 
