ON THE TRIGONOMETRY OF THE PARABOLA. 



11 



•whose amplitude is x* At the points V, B, C draw tangents to the parabola; 

 they will I'orm a triangle circumscribing the parabola, whose sides represent 

 the semi-ordinates of the conjugate arcs VB, VC, VD. 



XIII. We know that the circle circumscribing this triangle passes through 

 the focus of the parabola. Now 



VT=»?tan0, VT'=»itanx» T'A=m tan ^secx» TA=m tan^sec^; 

 hence 



T'A + TA=m(tan ^sec x+ tan xsec<|)), 

 therefore 



mtanw=T'A + TA. 



When VB, VC together constitute a focal arc, the angle TAT' is a right 

 angle. 



The diameter of this circle is m sec ^ sec x- 



The demonstration of these properties follows obviously from the figure. 



XIV. It may be convenient, by a simple geometrical illustration, to show 

 the magnitude of the functions sec(0-'-x) and tan (^-^-x)- 



Let SV=»J, ASV=x, BSV=0, the line AB being at right angles to SV. 

 Through the three points ABS describe a circle. Draw the diameter SC, 

 and join the point C with A and B. Let fall the perpendicular CT. 



Fig. 4. 



Then »Jsec(0-Lx)=SC + CT, and m tan (0-i-x)=AC + CB. 

 Moreover also it follows, since sec(0-J-x) + tan (0-i-x)=(seC(^+ tan^) 

 (sec X + tanx), as has been established in (6) of (III.), that 



»i(SC + CT+AC + CB)=(SB+BV)(AS+AV). . . (13) 



Of this theorem it is easy to give an independent geometrical demonstration. 

 We have manifestly also 



CT(SC + m+SA+SB) = (AC + AT)(BC + BT). . . (14) 



XV. Let w be the conjugate amplitude of w and \p, while w is the conjugate 

 amplitude, as before, of and x- Then as 



fsec wc?w = i sec w £?a» + V sec ^rfi//, and Tsec wc?w=f sec^rf^ + fsecx^^X' 



