DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 371 
Adding- these equations together, as 1 —n—j, we get 
\/\r n 2 
N 
Now the arc of the logarithmic hyperbola, as in (233.), is 
T A2 fcos^ipd^ 
r n 2 “I 
Ln“lJ- 
k v/B(A + C)J /I 
(c.) 
(d.) 
In this case, the coefficient 
¥ 
VB(A + C) 2 
value for this expression, given in (249.), m-=n ; hence 
^cos^ipd^ 
2 , as may be shown by putting in the general 
(e.) 
d(p 
; . . . (f.) 
Now (257.) gives = 
and the general value of ^ being — as in (256.), §=2/(1— w)^, /=2— w, and 
/— /^=/(l— w), since In — i^. 
The last equation may now be written, combining (e.) with it. 
Adding this equation to (c.), 
/<E>/ (1 +y)sin(p cosip V'T tanp-/! , tanp-ZT 
II 
k 
(287.) 
(288.) 
Now 
j iL j L 
Combining this value of O, with the preceding equation, we get 
yTMCOS^p 2C0S^p 1 
ll 
k 
= [‘an?'/ 1 -Jilp-v/l +jj^] + tan?yi [- 
N2 
LN 
]; (289. 
tanp VT 
and this latter term, in this case, may be reduced to—-' — 
ab 
mmj 
But, a and h being the semiaxes of the hyperbolic cylinder, (248.) gives^= ^ 
2 V ab k 
or in this case, as m=n. 
Vij j 
Now distance from the centre to the focus of an hyperbola, the squares 
of vvhose semiaxes are-«/» and'4a/», hence 
-J > 
[tan^x/T-Jd^x/1 4-j'J-^] 
% 1 
represents an arc of an hyperbola the squares of whose semiaxes arey«6 and'^o//. 
