90 REPORT — 1856. 



equal to the angle NSQ. The line ST will be =m sec d, the line ST^ 

 =msec(0-L0), the line ST^^ =m sec (0-^0-^0), &c., and we shall likewise 



have 



VT=wtan0, VT^=w/ tan (0-^-0), VT;,=>Mtan(0-L0-i-0), &c. 



This follows immediately from (6) of III. ; for any integral power of 

 (sec + tan 0) may be exhibited as a linear function of sec 6 + tan 9, 

 writing for 0-^0-1-0 ... &c., since 



sec(0-J-0-L0-i-0&c. to ?«0) + tan(0-L0J-0-^0&c. tow0)=(sec0 + tan0)". 



Hence the parabola enables us to give a graphical construction for the angle 

 (0_L0j-0 &c,) as the circle does for the angle (0 + + &c.). 

 I XXX. The analogous theorem in the circle may be developed as fol- 

 lows: In the circle SB A take the arcs 



AB=BB,=B,B,^=B„B,„ ... &c. =2^. 

 Let the diameter be D ; then 



SB=Dco9.&, SB,=Dcos23^, SB,(=Dcos3^ ... &c., 



and 



AB=Dsin:&, AB(=Dsin2&, AB^, = Dsin 3^ ... &c. 



Now as the lines in the second group are always at right angl es to those 

 in the first, and as such a change is denoted by the symbol V—l, we get 



SB + BA = D{cos&+ v/^siiid}, 



SB; + B^A=D{cos2^+ -v/^^sin25-} = D{cos&+ V-lsind}''; 

 SB;; + B,,A=D{cos3a+ \/^sin3^} = D{cos^+ V^ sin*f&c. 



SB^+B„A=DCcosn3'+ i/ITi sinw^] = D[cos.&+ \/^siua]"- 

 When the points B', B" fall below the line SA, the angle becomes negative, 



and we get 



SB' — B'A=cos.&— v/ — Isin^ 



SB" — B" A= cos 2^— V^ sin 2b= [cos ^— V^ sin b^- 



Therefore 



log(SB + BA)=log(cos*+V-lsin&)=^V-l. . . .(36) 



Let ^=1, then 



log[cos(l)+\/-lsin(l)]= V —1. 



Hence generally ^ \^^^ is the logarithm of the bent line whose extremities 

 are at S and A, and which meets the circle in the point B, ASB=.&. 



It is singular that the imaginary formulae in trigonometry have long been 

 discovered, while the corresponding real expressions have escaped notice. 

 Indeed it was long ago observed by Bernoulli, Lambert, and by others — the 

 remark has been repeated in almost every treatise on the subject since— 

 that the ordinates of an equilateral hyperbola might be expressed by real 

 exponentials, whose exponents are sectors of the hyperbola ; but the analogy, 

 bein'' illusory, never led to any useful results. And the analogy was illusory 

 from" this; that it so happens the length and area of a circle are expressed 

 by the same function, while the area of an equilateral hyperbola is a function 

 of an arc of a parabola, as will be shown further on. The true analogue of 

 the circle is the parabola. 



