358 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
From this expression we must eliminate the functions of <p. 
Now (73.) gives >/l= (189.) 
writing <p for j«-. 
Substituting this value of in the preceding expression, for which we put t, we get 
n\ . , 2 sina cosal 
te2|sm+ (190.) 
From this equation we must eliminate sin^, cosip. 
If we solve the preceding equation (189.) we shall obtain the resulting expressions 
2sin"p=l— VI, cos-^+i,sm^} 
2 cos^p= 1 + cos4>—i, sin^-^/ J 
Multiplying these equations together, and recollecting that 1^=1— f sin^-v//, we find 
4 cos^p sin^p=sin^4'[I/+2 cos-4/+*f cos^-v^] (192») 
Now the second member of this equation is a perfect square, 
whence 2 sinp cosp=sin'4/[ '/T^+z^cos-vl/] (193.) 
Substituting this value of 2sinpcosp in (190.), we get 
<=| Sin4-[l - A + yos'|,j 
As w=l —j, and equation (187.) may now be written 
« /I +i \ Fj , /r sinP cosP , , , 
"^=2 
■».T i ® (2— «) a (1+/) , , 2 
Now as A=o ' /v=— =o — and 
2 VI— n 2 V} ' ' 1+^’ 
we get ultimately 22=A(d'vj/ A (196.) 
J 'Vl — rsin^p 
The second term of the last member of this equation is evidently the common 
expression for a portion of a tangent to a plane ellipse between the point of contact 
and the foot of a perpendicular on it from the centre ; while Ajd-i^ Vj, or 
Ajd4/ is the expression for the arc of a plane ellipse whose semitrans- 
verse axis is A, and eccentricity i,. 
When the function is complete, P =2 and ■4 >=t. See (183.) 
Hence as Jd-v^/ Vl—2^o'‘d’4^ Vl„ 
2=Apd^^.^I, (197.) 
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