ON THE TRIGONOMETRY OF THE PARABOLA. 79 



Now adding the three latter equations together, and subtracting the sum from 

 the former, the logarithms disappear, for 



.og(L±_!^) + log(^=) + >og3= log[3 . (l±f^) (£^)] 



= log(6 + 3'/5); Cl8) 



consequently 



U(m.u)—U(m.<p)-n(m.x)—Tl(m.^P) 



sincetan(rf.-^x)=2, tan (<^-L4/)=^±i^, and tan (x-^>//)= ]^±AzI^ 



6 6 



XVII. Let, in the preceding formula (16), ^=x='/') ^"d we shall have 

 n(m.(D) — 3n(m.(^)=2mtanX0-J-x) = 16mtan^0sec^^. 



We are thus enabled to assign the difference between an arc of a parabola 

 whose amplitude is w = (^-^0-^0) and three times another arc. 

 If in (ct) (III.) we make ^=j^=;p, 



tanw=4tan^0 + 3tan^ (20) 



Introduce into this expression the imaginary transformation 

 tan(^= V—i sin 6, change -■- into +, 



and we shall get sin 30=— 4 sin^9 + 3sin d, which is the known formula for 

 the trisection of a circular arc. (20) may therefore be taken as the formula 

 which gives the trisection of an arc of a parabola. 



XVIII. The following illustration of the triplication of the arc of a para- 

 bola may be given : — 



Take the arcs whose ordinates Y and y are 4w2 and m respectively. Let 

 w and (j) be the amplitudes which correspond to these ordinates ; then as 



Y=2mtan o3=4;m, tana)=2, secw=\/'5; 

 and as , ^- 



y=2mtan0=m, tan^=-, sec0=__?* 



Now these values of tan uj and tan satisfy the equation of condition (20), 

 namely 



4 tan' ^ + 3 tan 0= tan w. 

 But 



n(m . tan-1 2)=m2 V's+m log (2+ V^S), 



and / , _ 1\ 1 V'5 , , fl + ^5\. 



ll\m .t&n-^-^J = m^~^ + m.]ogy — ^— I , 



and three times this arc is 



3n(m . tan-^ ^\ = }njV5 + mlog(2+ -/s), 



since /H-V^^\' /- 



Subtracting this latter equation from the former, the logarithms disappear, 

 and we get 



n(m.tan-i2)-3n(m.tan-i^j = ^^i^ = I6OTtan^0sec'f . (21) 



