DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 407 
In (186.) we have shown that the rectification of a plane ellipse whose semiaxes are 
a and h, may be reduced to the rectification of another plane ellipse whose semiaxes 
ffp are given by the equations / 2 ^=a+Z>, ab, of which the eccentricity is less 
than that of the former, a + 6 is that portion of the tangent, drawn through the point 
of maximum division, which lies between the axes; and \/a6 is the perpendicular 
from the centre on it. 
VVe have shown in (63.) and (74.)? tli^t If?* and are connected by the equation 
tan(-<p — <p)=j tamp; while i and are so related, that 
we shall have 
_1— V'l—i 
l-i 
1 +/ 
c dip _ (i+?,)r d\t> (1 +?'/ )rd\t/ 
J a/ 1 — sin®(p 2 J A^i— ^ J'^1, 
Let us now introduce this suggested transformation into the elliptic integral of 
the third order, circular form and negative parameter. In (191.) we found 
Now 
2 sin^9=l+^^sin-•4/ — cosa|/\/I^. 
r _ C d 
Jma/I J[ 
! [1 — m sin®<p] I — sin^ip" 
Or replacing (p by its equivalent functions in 4/, and recollecting that m—n-\-mn=i^, 
since m and n are conjugate parameters, we shall find 
d^I; 
Jm 1 ^ ^ — m — 
sin^\I/ + m cos\I; I^] \/\’ 
(394.) 
We may eliminate the radical m cos-vl^v/l; from the denominator of this expression, 
by treating it as the sum of two terms. 
Multiplying and dividing the function by their difference, since 1 + 
C/ 
— m — mi, sin^vf; — m cos\I/ 1 ,] 
(395.) 
Now it is truly remarkable that whether the parameter of the original function we 
start from be positive or negative, the parameter of the first derived integral will 
always be positive. Indeed it is necessary that this should be the case, because the 
parameters of the derived functions, increasing or diminishing as they do, must at 
length pass from between the limits 1 and Should they do so, the integral would 
be no longer of the circular form, but of the logarithmic. Now we cannot pass from one 
of these forms to the other by any but an imaginary transformation. This objection 
does not hold when the parameter is positive, because the limits of the positive para- 
meter are 0 and co. It is, too, worthy of remark, that the first derived parameter is 
always the same, whether we transform from positive or negative parameters. Write 
is the first derived parameter. 
mn 
3 G 
( 396 .) 
MDCCCLTI. 
