DR. BOOTH ON THE GEOMETRICAI. PROPERTIES OF ELLIPTIC INTEGRALS. 411 
Or substituting- the values of m^ and w, in terms of m and n, and therefore of a and b, 
a + b , 2 '</ ab 
\ab’ 
*+Ii 
(412.) 
When the radius of the sphere is infinite, or the derived curve is a plane ellipse, 
a=a-\-b, bi=2y/ab, as in LXXVII. 
When m—n = i\ or when the given curve is a spherical parabola, the 
derived curve will also be a spherical parabola. Hence all the curves of the series 
will be spherical parabolas. 
If we take the corresponding integral of the third order with a reciprocal para- 
meter /, such that lm=i^, and derive by the foregoing process the first derived 
function of the third order, we shall find the parameter of this function to be 
positive also, and reciprocal to so that 
Hence, if we deduce a series of derived functions from two primitive functions of 
the third order and circular form, having either positive or negative reciprocal para- 
meters, the parameters of all the derived functions will he jjositive, 
and reciprocal in pairs, so that 
LXXX. We may apply the same method of proceeding to the logarithmic ellipse, 
or to the logarithmic integral of the third order, 
d(p 
— msin^ip) ■v/l — sin^^’ 
in which i*>m. 
If on this function we perform the operations effected on the similar integral in 
(394.), we shall have, after like reductions. 
C 
(1 + */) 1 
pd4/[2 — m — mil sin^vj/ — m cos\|/V^kl 
iMv/l 
t 
■ 4(1 —7W)^ 
) [1 — sin^4/] 
(413.) 
We must recollect that 
M=: I — m sin^<p, M,= 1 — m^sin^-v^, 1= 1 — f^sin^, I,= 1 — if sin®'v^, and/w^=^j^^. (414.) 
We may reduce this expression. 
The numerator may be put under the form 
Novr 
2 — \ sin^-4/— 1 } —m cos-\p\/ 1^. 
^ mi, (n—m) , mi, i* 
2 — m—— = and — We have also 
m, n ^ m, n , 
1 
1+/ 
Hence, making the necessary transformations, 
2^ 
m 
0| 
p df {n — m) \/iiC d^/ * 
’ d4/ \/ii{ 
[’cos^J/d^/ 
J 
Im -\/T 'oin i J^/ -v^k J 
^ J e J 
1 M, 
If into this expression we introduce the relation given in (74.), 
we shall have 2 
I JM, V\,rnnJ ./f i J M, 
Mv^I 
(415.) 
