^4 REPORT — 1856. 



It is not difficult to show that 



tan w sec w — tan (p sec ^ — tan ^ secx=2 tan w tan (j> tan )^. 



Substitute for tanw, sec w, their values given in (1) and (2). Write 

 (sec^ — tan" 0) and (sec^x~t^°"x) ^^r 1, the coefficient of tan sec ^ and 

 tan X sec x io the preceding expression, and we shall obtain the foregoing 

 result. 



VI. Let y, J/', y" be the ordinates on the axis of the parabola of the ex- 

 tremities of the arcs n.(m . w), n(m . (p), and U(m . x)- Then y=2m tan w, 



y =2 m tan 0, y"=2m tan x- Therefore 2m tan w tan ^ tan x = ^MJ^. 



We have therefore the following theorem : — 



T/ie algebraic sum of the three conjugate arcs of a parabola, measured 

 from the vertex, is equal to the product of the ordinates of their extremities 

 divided by tlie square of the semiparameier. 



To exemplify the preceding theorem. Let 



1 Vl 



tanw=2, t3M<p=-^, tanx= -o— ' 



then /- V5 ^„„ _ 3 



secw=v5, sec ^=—^5 secx— -3-; 



and these values satisfy the fundamental equation of condition, 



tan w= tan sec x+ tan x sec 0. 

 Now 



n(m . w)=»n2 V'S +«» log (2+ v/5) 



n (m . 0) = m -^ +m log(^ — ^ j 



n(m.x)=wi-^ + mlog(^ 2~/* 



Hence, since log (2+ V~5)= log( i±^ j + log ( ^"'" )> we shall have 



n(m.w)— n(»w.0)— n(m.x)=»iV5; .... (9) 



and m, V5'=2m tan w tan <p tan x- 



VII. If we call an arc measured from the vertex of a parabola an apsidal 

 arc, to distinguish it from an arc taken anywhere along the parabola, the pre- 

 ceding theorem will enable us to express an arc of a parabola, taken any- 

 where along the curve, as the sum or difference of an apsidal arc and a right 

 line. 



Thus, let VCD be a parabola, S its focus, and V its vertex. Let 



VB=n(»i.0), VC = n(m.x), VD=n(»i.w), and let^^^^=/«. Then (8) 



shows that the parabolic arc (VC + VB) = arc VD— A ; and the parabolic arc 

 VD-VB=BD=VC+/«. 



VIII. When the arcs U(m . (p) and n (m . x) together constitute a focal arc, 



or an arc whose chord passes through the focus, 0-|-x= -, and h is the ordi- 

 nate of the arc VD. Accordingly we derive the following theorem : — 



Any focal arc of a parabola is equal to the difference beticeen the conjugate 

 apsidal arc and its ordinate. 



