86 REPORT — 1856. 



This is easily shown. 



Let e^-i-e^=e. Then sec(e^-^e;)^sece=sec°ej+ tan-e^, and 

 tan (ey-^ey)= tan e=iisece^tan e^. 

 Therefore sec ( e^ -^ e^) + tan (e^-'-e^)=sece+tane=e= 



sec* e^ + tan^ e^ + 2 sec e^ tan e, = (sec e^ + tan e^)". 

 Hence sec6(+ tan e;= V^e (33) 



2 ' 

 taii(e-^e)=tel^, 8ec(e^e)=^l±^; 



tan(e-Le-^6)= ^ ~^ -, sec(e-^e-^e)= ^ "'^^ ; 

 tan(€-^e-L- to n terms)= ' sec(e-Le to n terms)=^ — -S — 



Therefore 2 sec e tan e= tan (e + e) 



2sec(e-Le) tan (e-Le) = tan (e -^ e ->- e -^ e), 

 and generally • 



2 sec(e-^e-'- to « terms) tan (e-^e-^ to « terms) = 



tan (e-^e-^e-'-e-^ to 2w terms). 



Now 2 sec (e -■- e -i- to n terms) tan (e-'-e-^ ton terms) is the portion of the 

 tangent to the curve intercepted between the axis of the parabola and the 

 point of contact whose amplitude, or the angle it makes with the ordinate is 

 (e-L-e-^ to n terms;, while tan (e-^e-^e-^e-^ to 2n terms) is half the ordi- 

 nate of that point of the curve whose amplitude is (e-^e-^e-^e to 'in terms). 

 Hence we derive this very general theorem : — 



That if tivo points be ttiken on a parabola such that the intercept of the 

 tangent to the one between the point of contact and. the axis shall be equal to 

 one-half the ordinate to the other, the amplitudes of the two points tvill be 

 (e-i-e-L to « terms) and (e-^e-^e-^e to 2n terms) resjjectively. 



This theorem suggests a simple method of graphically finding a parabolic 

 arc whose amplitude shall be the duplicate of the amplitude of a given arc. 

 Let P be the point on the parabola whose amplitude is given. Draw the 

 tangent PQ meeting the axis in Q. Erect VT at the vertex = PQ. Through 

 T draw the tangent TP', the amplitude of the arc VP' will be the duplicate 

 of the amplitude of the arc VP, or {d-^Q-^ to n terms) and {B-^d-^ to 2n 

 terms) will be the amplitudes of VP and VP' respectively. Vv'e may there- 

 fore conclude that in the circle 



2cos(<i + 0+ to n terms) sin (9 + 9+ to n terms) = 



sin(0 + e + + to 2w terms). 



XXVIL In the trigonometry of the circle, the sine of the arc, which is x 

 times the radius, is given by the formula 



x^ x^ x' . 



sm x^=x • -I , &c., 



123 ^ 12345 1234567 



and the cosine of the same arc by the formula 

 cosa;=l- — + 



12 ' 1234 123456 



