252 
Mr. Woodhouse on the Integration 
axes 1 and |/(i — e*) ; consequently, the arc of one ellipse may 
be represented by the arcs of other ellipses whose excentricities- 
vary according to a given law ; thus, 
and • i= r . fir = T" 
Consequently, 
e' #'—( 1+0/+/ — -Ujp- . a''—/' ( 1 + e") +/" }, 
3 +<* 
7T-/'- 
i — d' 
;/"• 
ore' «'_(!+«')/+ 2 . (I+e , )V Tilr+inv 
which equation, calling sin. 20', w" sin. 40", agrees with the 
equation 
2 .( 1 +0 E"== (E+c.sin. <p) + 2<r.sin. <p, 
given by Legendre, Mem. de VAcademie , 178b, page 657. 
If a: = 1, «'= o, and w"= o ; consequently, 
*/'(»)+ (6) 
Putting * = ^ we have (see page 246) df= dv —^-^r • 
-i=*v/(S)- 7h+ , ‘)*'+T ; C 1 +'*) i T^+ 
s' 
1 +'6 d I +'b" \r \ i+z' z ]' 
Similarly, /= - 1 * ( I+ »6) z"+^ (i+" 4 ).- + 
+ « 
Hence, exterminating there results an equation between/, /,'/, 
and certain functions of v. 
*'= 
V* 
-~ } substitute these quantities, and 
