DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 359 
2 therefore, in this case, is equal to a quadrant of the plane ellipse whose principal 
semiaxis A, and eccentricity i, are given by the equations 
A=\/ and 
. 1 — '/l— 
''“l + 
(198.*) 
To distinguish this variety of the curve, we may call it the circular logarithmic 
ellipse, as it is a section of a circular cylinder. Accordingly, in the two forms of 
the third order, when the conjugate parameters are equal, or m=n, the representative 
curves of those forms become the spherical parabola, and the circular logarithmic 
ellipse. 
This is Case V. in the Table, p. 316. The results of the preceding investigation will 
reappear in the demonstration of the theorem, that quadrants of the spherical or 
logarithmic ellipse may be expressed by the help of integrals of the first and second 
orders. 
XLIV. It is not difficult to show that this particular case of the logarithmic form, 
when the parameters m and n are equal, represents the curve of intersection of a 
circular cylinder, by a paraboloid whose principal sections are unequal. 
2 2 
Let and ^+| 7-=:22 (199.) 
be the equations of the circular cylinder and of the paraboloid. 
Assume a:=acos0, 3/=asin0 
Then ‘Iz^cCA 
and 
Hence 
( COS^d 1 
sin^^ 
[ A “1 
r k' 
Ax 
Td'- 
fl sin0, ^=acos0, = sin9cos0. 
sin'Ocos^e] 
1 1 
( 200 .) 
( 201 .) 
( 202 .) 
Now we may reduce this expression by two different methods to the form of an 
elliptic integral. 
By the first method, eliminating cos^0, this expression becomes 
sin<0 (203.) 
We may, as in (124.), reduce this expression to the form of a product of two 
quadratic factors, 
(A+B sin^6)(C — B sin^(9)=:AC+B(C— A) sin*0— sin‘‘0. . . . (204.) 
Comparing this expression with the preceding, 
AC=a\ C-A=a^Q--^ or C=A+B, and AC=A^+AB=tf^ (205.) 
* Professor Stokes of Cambridge has pointed out to me, that this curve, like the plane ellipse, when the 
cylinder is developed on a plane, becomes a curve of sines. 
MDCCCLII. 3 A 
