DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 405 
T being the normal angle of the tangent parabolic arc to the logarithmic ellipse, 
this, evidently, will be a maximum when the parabolic arc is a maximum. Put the 
dx • • 1 
differential coefficient This gives, as before, tan<p=-^. Substituting this ex- 
pression in (a.), we get 
• - mn 
sinr=- 
(l+» 
(392.) 
We shall find the importance of this expression in the next section. 
From (392.) we derive 
tanV: 
mn 
4 
Now 
Whence we get tanr= 
(l+j')^=2-l-^" — /^=2-l-!^* — m — n-\-mn. Hence as 
J=\/ (1— m)(l— /?.), 1— w]^ 
V mn 
Multiply this equation, numerator and de- 
V 1 —m-\- V \—n 
nominator by s / ] — m —\/ 1 — w, and the last expression will become 
tanT= 
\/ mn \ — m ^ mn v/ 1 — j 
n—m n—m 
In (171-) we found for the semiaxes of the cylinder, whose intersection with the 
b i/ mn v/ 1 — n 
paraboloid is the logarithmic ellipse, m 
n—m 
n — m 
Hence 
tanr= 
',—b 
(393.) 
This gives a simple expression for the tangent of the maximum parabolic arc, ana- 
logous to (385.) and (391.). We have only to take in the parabola, whose semi- 
parameter is A-, an arc whose ordinate is a — b, to determine the maximum protangent 
parabolic arc. 
The value tan<p=^^, which fixes the position and magnitude of the maximum 
protangent arc to the logarithmic ellipse, renders tan^X=^. For (150.) gives 
tanV=— tan^X. But (152.) gives — and 
tan^A 
hence tan*(p=^j . If we now make 
^ 1 —m 
1 
tan ?)=:-=:- 
. 2-1 / 1 — ^ ® 
tamX=\/ = rj 
V l—n b 
J V {\—n){\—m) 
as we may infer from (171 •)■ Now substituting this value of tan^X in (155.), we 
shall get 
tanT=- 
Again, if we differentiate the values of x,y, z given in (158.), the coordinates of the 
extremity of the arc measured from the minor axis, and substitute them in the general 
