ON THE TRIGONOMETRY OF THE PARABOLA. 75 



Fig. 2. 



The relation between the amplitudes ^=r|—x) and w in this case is 



siven bv the equation sin 2<b= ^^^^'^ . Thus when the focal chord makes 

 ° ^ ^ 1 — cos W 



an angle of 30° with the axis, we get cos w=i, or y=\Om. Here, therefore, 

 the ordinate of the conjugate arc is ten times the modulus. 

 IX. When ^=X' (8) is changed into 



n(OT.w)— 2n(m.^)=2mtanwtan"^; .... (10) 



or as tan w=2 tan </> sec 0, see (?j) of III., 



n(m.w) — 2n(m.0)=4mtan^<^sec^ (11) 



Let ^=45, then n(»».-^ is the arc of the parabola intercepted between 



the vertex and the focal ordinate; and as sec w=sec(^-J-0)=sec20 + tan^0, 

 we shall have, since tan 0=1 and sec 0= V^, sec w=3 ; therefore 



n(m.sec-i.3)-2nf»»-y=4'mV2. 



Now as sec a)=3, tan w=2 »/% and the ordinate Y=4m'/2^we may there- 

 fore conclude that the parabolic arc, whose ordinate is ^mV% diminished by 

 this ordinate, is equal to the arcs of the parabola between the focal ordinate 

 produced both ways, and the vertex. 



X, It is easy to give an independent proof of this particular case without 

 the help of the preceding theory. 



The length of the parabolic arc whose amplitude is 45° will be found by 

 the usual formula to be 



n(»w.^j=W'/2+OTlog(l+A/2); 



