360 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
Let us now, as in (126.), assume ; (206.) 
and following the steps there indicated, we shall have 
2=a\ — L — (^+^) J (207.) 
Ir, B . , A B(2A + B) . ^ 
J L' - aTb V 1 - - (a+ ' b A 
an expression of the same form as (127.)- 
^ B B(2A + B) .2 
A + B""^’ (A+B)^ “■* ’ (208.) 
A "1 
therefore 1-^=ATB’ l~*'=(A+^n (209.) 
Hence 1 —n= — P, or n=m i 
If we develope this integral by the method indicated in XXXVI., the coefficient 
^ d(3 
of the integral „ , / — in the result will be 0, and the re- 
duced integral will become, as 
^=n, B=y:^, andB=a’(j,-j) (210.) 
C^n.) 
Let z' and z" be the altitudes of the points above the plane of ut/, in which the 
principal sections of the paraboloid meet the circular cylinder. Then z" — z' is the 
height or thickness of the zone of the cylinder on which the curve is traced. 
Now a^=2kz', a^=2k'z", whence 2"— — 
Let this altitude or thickness of the zone be put h, and we shall have 
Hence the arc of this species of logarithmic ellipse may be expressed by integrals 
of the first and second orders. 
It is not a little remarkable that whether the integrals of the third order be circular 
or logarithmic, or, looking to their geometrical origin, spherical or parabolic, when 
the conjugate parameters are equal, or m=n, we may express the arcs of the hyper- 
conic sections thus represented, in terms of integrals of the first and second orders 
only ; the integral of the third order being in this case eliminated. 
If we now resume equation (202.) and make 
20— o+A 
(213.) 
