384 DR. BOOTH ON THE GEOxVIETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
2 -/ 1— 2 
-Jo 
+(I.) 
cot0 (Ie)J 
constant^ 
(332.) 
writing H for i 
u» 
or in the ordinary notation, 
■ dip 
Vt 
fd0\/(lo)' 
d9 
(333.) 
H=F,E,(0)-E,F,(0). 
When we require to determine the constant, we must not suppose 6=0, for this 
would render n=0, and so change the nature of the curve. Neither should we be 
justified in making i=0, (as some writers do), for this would be to violate the original 
supposition — and all the conclusions derived from it — namely, that { is constant, and 
less than 1. Moreover, since m-\-n — mn=P=0, on this hypothesis, m-\-7i=mn ; or ?n 
and n would each be greater than 1, which is inconsistent with the possible values of 
those quantities. 
We have now to determine the value of the constant. In these investigations we 
have all along supposed w > /w. The least value n can have is n=m. Were we 
to suppose n to be less than m, it would be nothing more than to write m for n, since 
m and n are connected by the equation m-\-n — mn=i^. Hence if m is not equal to 
n, one of them must be the greater, and this one we agree to call n, writing m for 
the lesser. To determine the constant, let us assume n=m. 
Now n=i^ sin^0, and n, when equal to m, is = 1 — \/ 1 — (1^)=: 1 — s\n^6 =\/ 1 — i^ 
cot^6=\/l — and tan6=0y. Hence the coefficient of H in the last equation, 
■/(Ta cot0 ^ ^ 
becomes 0, since in this case cot0=..^l — ; and as n=m, the curve is 
the circular logarithmic ellipse. See Art. XLIII. 
The last equation now becomes 
— constant (334.) 
Now if we turn to (I76-)5 shall find this, without the constant, to be the ex- 
pression for the quadrant of a circular logarithmic ellipse, or the curve in which a 
circular cylinder, the radius of whose base is a, intersects at an infinite distance a 
paraboloid indefinitely attenuated. Hence the constant is 0. 
To determine the value of the above integral, when 0=^- 
In this case, as H=FjEi— E;Fj, H=0. And as cot0=O, and v^l 9 =\/l — the 
equation (332.) will assume the form 
How are we to interpret this expression ? 
0 y- 
+Vr 
(335.) 
