80 REPORT— 1856. 



Now as the radius of curvature R is equal to the cube of the normal divided 



by the square of the semiparameter, 11= — ^, since N = 2m sec w. We 



4 

 have therefore the following theorem : 



The arc of the parabola ivhose ordinate is equal to im, or to the abscissa, 

 diminished by the radius of curvature of its extremity, is equal to three times 

 the arc ivhose ordinate is m, or one fourth that of the former arc. 



It is evident that the cliord of the greater arc is inclined by an angle of 45 

 to the axis, or the ordinate is equal to the abscissa, while in the lesser arc the 

 ordinate is four times the abscissa. 



This is the point on the parabola up to which the ordinate is greater than 

 the abscissa; beyond this point it is less than the abscissa. 



XIX. Another example of the triplication of the arc of a parabola, or of 

 finding an arc, which, diminished by an algebraic quantity, shall be equal to 

 three times another arc, may be given. 



Let 



3 

 tan0=— , tanw=18, 



-/Is _ /— 

 sec(fi= , sec w = 5 V 13. 



These values satisfy the equation of condition, 



4 tan^ + 3 tan 0= tan w. 

 Hence 



n(»i.tan-i.l8)=OT90. VIs + w log (18 + 5 VlS) 



n(»i.tan-»-j =m—^ +rn\og\-^ j; 



and three times this arc is 



Sn (wi . tan- 1 1) =^^^ + m log ( 1 8 + 5 ^TS), 



(?±^)^=18 + 5^T3. 

 Therefore subtracting the latter equation from the former, 

 n(/w.tan-q8)-3n('m . tan-i = »»!£i^^=16m(^|)'(^) . (22) 



XX. To find the arc of a parabola which shall differ from 7i times a given 

 arc by an algebraic quantity, may be thus investigated : — 



Let be the amplitude of tlie given arc, then 



n(m.(^)=msec(/) tan + ??i log (sec ^ + tan <f), 



and n times this arc is 



7iW{m . (p)=nm sec (p tan f+mlog (sec + tan (p)". 

 Let -•-(/) -^0 -J- to w terms =il>, then 



n(m.*)=msec$tan $ + m log (sec 4> + tan $). 



since 



