DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 415 
therefore 
4 
and as ^Yq:^=(l + ^)^ the expression will finally become 
n-m = {\-\-j)\\-m)s/n,, hence (430.) 
If now we add together (420.), (425.), (426.), (428.) and (429.), we shall have, 
..... , (n-m) 
dividing by 
(431.) 
Let us now take the logarithmic ellipse whose equation contains n^, instead 
of m, n, i and (p, we shall have from (163.), 
22' my mp, ^ ^ l—mp \ . C dv|/ 
k,~'~ n-m, V "*/ y V VJ, 
r - 21 "^ ( 432 .) 
n, — m,J ‘ ' m,\n—m,J^ ' Jcos^t, '■ ^ 
If we now subtract these equations one from the other, combining together like 
integrals, the integral of the third order will vanish and we shall have, 
k k n,(7i,-m;)(l-m,)Lj J ^ Jcos^t Jcos^r, V J 
Hence, as we may express an arc of a plane ellipse by an arc of a derived ellipse, an 
integral of the first order, and a right line — a known theorem — so we may extend 
this analogy and express an arc of a logarithmic ellipse by an arc of a derived loga- 
rithmic ellipse, by functions of the first and second orders, by an arc of a parabola 
and by a right line. The relations between the moduli and amplitudes are the same 
in both cases, 
1 —j 
tan(-4/-<p)=:j tan^. 
Let m,„ n,^, i,„ be derived from m„ n„ y, -v^, by the same law as these latter are 
derived from m, n, i, (p, namely, 
1—7 /IV- nfin 
h=i:ry tan(-4/-(p)=:jtan(p, = 
n 
= [■ 
-v/l— m— V^l 
] 2 
. 
V 1 — m+ V 1 ■ 
and derive an arc of a third logarithmic ellipse, we shall have, putting A, B, C, D for 
the coefficients of the integrals, and 11 for the parabolic arc. 
f'-f=4’'f'^l+Bj*^_CT+Dn, 
T-f=A'Jd+A/i:+Bj’^;c-'T,+D'n, 
3 H 
MDCCCLII. 
