368 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
Now as m+n-mn=P, the first factor of the numerator becomes (l-u)(1 -nisin'?) 
= (1— w)M, and therefore 
/TZFrfdt. fd. I 
V /(/— l)Ljcoso JcosTj \ I J} LN^/I 
Substituting the second member of this equation for the last in (268.), we find 
/■ df , f df (if fLl-i»sin> ] .... (275.) 
LNv'I • • • 
Now, since we have assumed in (269.) 
LN 
. , x'tanip 
siny'= 7 =^, cosV — t — q-s hence 
'I COS^ip’ 
do' _ V^ Tl-»^sin^<P]d? . / 07 « ^ 
LN^/I , . • • t-/ V 
cosu 
r dip ( 277 .) 
and consequently ]F7 T'^Jn\/T '/xjcoso' 
This formula is usually written 
f if .. 
J[l-nsm’fi]-/l-c*sin> sinVjv l-c’sin>f 
^■v/ atanip^^ 
We have thus shown that from the comparison of two expressions for the same 
arc of the logarithmic hyperbola, we may derive the well-known equation which 
connects two elliptic integrals of the third order, and of the logarithmic form, whose 
parameters are reciprocal*. 
Hence also it follows that if v, r, and v' are three normal angles, which normals 
to a parabola make with the axis, and if their angles are connected by the equations 
COS 
cos'V = 
cosV=i 
ML . I ff*' /T 
V = — o:, Siny=\/ — tampv^I, 
cos (5 V ^ 
MN 
1 ’ 
LN 
COS^ip’ 
sinr= 
mn sinip cos<p 
a/I 
. (2-9.) 
we shall have 
S'«"'='\/^( 
^tan® 
^do 
^dr 
. . . . (280.) 
Jcoso 
jcoso ' 
JCOST 
* We might by the aid of the imaginary transformation sin^= a/- 1 tantf/ have passed from this theorem, 
connecting integrals with reciprocal parameters, to the corresponding theorem in the circular form. It seems 
better to give an independent proof of this theorem by the method of differentiating under the sign of integra- 
tion as we shall do further on. Although these theorems have algebraically the same form, their geometncal 
significations are entirely different. In the logarithmic form, the theorem results from the comparison of two 
expressions for the same arc of the logarithmic hyperbola. But in the circular form, the theorem represents 
the sum of the arcs of two different spherical conic sections described on the same cylinder by two concentnc 
spheres, or on the same sphere by two cylinders having their axes coincident. 
