254 Mr. Woodhouse on the Integration 
It is plain that, by a similar method, we may deduce an equation 
between/ ( 1 ), /'( 1 ), andJ.-pt-. 
Legendre puts c"= therefore /answers to E" in his equation. 
From what has preceded it appears, that the forms 
/= y ■ 1 •! 7 ,' +:i + !i " r- J-h. 
f—j 
b* 
4 
6 . 'P 'CL 
{ f • (»+'*) *+ ** + &c. } 
e u 
T+7 
•'•(t+W+C)). 
e" (i — e")'. u" . 3 + e" 
(I-O/" 
2 . (i+f?') (1+0* 1 2 . (1+0 ( 
; ft u— e 
*■+«')*•' 2 -0 + 
O (i+O a 
&c. 
are parts of the same method of computation, differently ex- 
pressed. It also appears, how certain analytical artifices of com- 
putation, translated into geometrical language, become curious 
properties of curves. 
Fagnani’s theorem, as it is called, may be deduced from the 
form for the transformation of f; thus, taking the simplest case, 
0—0 du< 
2 
c r+7+?- d J’’ 
df [l+e’) . du’ ■ 
when u' is at its maximum, ( l) x = ) 
•••/(•;•= OH=f (’+"') - 
and/(i) (*=i) = — ^• 2 Ftr + -Tj4 > - 
Consequently, 2 / — -/( x) = -2A (l+^') = (l — 4*) • — ^ 
"(»-*•) ITT = »-*; 
