1,58 Mr. Wood house’s Demonstration of a Theorem, 8 cc . 
4. The length of the curve cannot be algebraically expressed ; 
but it may be exhibited by means of the rectification of an 
(r-zxy 
elliptic arc; + 
r + x 
(r-x] 
= ~/A 
If c and t are semiaxes of an ellipse, v abscissa measured from 
vertex, arc (A) = v 
+ 
1 , let v — t 
tv x. 
,2 tv — v? 1 i 1 
then 4 ~ HL If + . 
2 vyv \ r’J^V I t' — tn'rx J 
= C — ^ (semiaxes t, and c, abs. t — mtV. x). Compare this 
form with -=v/(A and f = r, »*= «* = sr .-. 
= — \/ r x A (semiaxes v/ 2 r, \/ r, abs. — v/ a:) -f* corr. (C) 
when ,r = o, length of curve = o C = s/ r x quad. el-, 
lipse (Q). Hence, length = \/"r (Q — A) when x — r=\/ r *Q 
= Q' (Q' being quadrant ellipse of which the semiaxes are r, 
r s/ 2) ; hence, if an ellipse be described, of which the semiaxes 
are the radius (r) of a great circle of the sphere, and the side 
of a square inscribed in that great circle, then the length of the 
curve line which is the intersection of the cylinder with the 
surface of the hemisphere, is equal half the periphery of the 
ellipse. 
• The length of the curve may be as commodiously expressed in terms of g ; fpr, 
since x — g cos. 6, y ~ g sin. 0 , z = vV— f V **+>* + = \/ A + V) g‘ 0 ‘ 

but 0 — ■— — therefore V -f f -j- z ' 1 — g V ' - ; whence the integral, 
by means of the rectification of an ellipse, as before. 
