76 REPORT — 1856. 



and twice this arc is 



2n(m.^)=?w2V'2 + ?wlog(3+2\/2);since(l+\/2)2=3 + 2V2. 



The parabolic arc whose amplitude is sec-^ 3, is found in like manner to be 



n(m.sec-i3)=»i3.2'/2 + mlog(3+'/2). 

 Subtracting the former equation from the latter, 



n(m.sec-i3)— 2n(«».^)=4m\/2. 

 Now the ordinate Y of the parabolic arc whose amplitude is sec~' 3 is equal to 



2m.2'/2=4'm'/2, 

 therefore jj(m . sec"' 3)-m{m.^\ =Y. 



It is easily shown that 4»j \^2 is the radius of curvature of the extremity of 

 the arc whose amplitude is 45°. 



XI. To find a parabolic arc which shall differ from twice another parabolic 

 arc by an algebraic quantity, may be thus exemplified. 



Let tan = 2, tan w = 4 V^S, 



sec = V 5, sec w= 9, 



then n(m .sec-i 9)=»w36'V/5 + »ilog (9 + 4 i^S) 



2n(wJtan-i2)=2wj 2 4/5 + m log (2+ ^^5)% 



Consequently, since (2+ '/5/=9 + 4V'5, 



n(?w.sec-'9)— 2n(»J.tan-i2)=;w32\/5=2mtanwtan^0. . (12) 



XII. We may in all cases represent by a simple geometrical construction the 

 ordinates of the conjugate parabolic arcs, whose amplitudes are <(>, Xj and w. 



Let BC be a parabola whose focus is S and whose vertex is V. Let 



Fig. 3. 



■VS=w; moreover, let VB be the arc whose amplitude is f, and VC the arc 



