CONIC SECTIONS. 



165 



Scholium. 



If, hi the case of the circle, we denote the angle 

 ACD by u, then I = a tan. u, \Z(a z -{-l') z= a sec. u = 



we have n = -, 

 e 



and 



COS. u 



-f- a = 



a (1 + cos. «) 



a (I — cos. it) 



cos. u 



\/(« 2 + n 



l' 1 = «' 



I" 1 = a* 



t" 



therefore, 



1 — cos. u 

 1 -f- cos. u 

 1 ~— cos. ^ w 

 1 -f- cos. ^ w 

 1 — cos. 4- u 



&c. 



1 -f cos. 



In this manner the quantities V % , t" 1 , &c. may be readily 

 formed from the cosines of the submultiples of the an- 

 gle u ; and these may be derived one from another by 



the formula cos. \ u =r ^J ( 9 )• A similar trans- 

 formation may be made in the series T in the case of 

 the hyperbola ; also the quantities i", I" 1 , &c. may in 

 this case be found from the trigonometrical tables, by 



considering, that if Sin. z= — , then l n =Ta.n. 1 \z, &c. 



( Scholium to last Prop. ) The series T, in this formu- 

 la, converges more rapidly than that in the last, the 

 terms approximating continually to those of a geome- 

 trical series, of Avhich the common ratio is -^ ; from 

 which it follows, that the sum of all the terms follow- 

 ing any term, is nearly ^V °f that term ; and this is 

 more nearly true, as the term is more advanced in the 

 series. 



By the same mode of deduction, other formulae for 

 the areas of a circle and hyperbola may be found, but 

 for these we refer the reader to a paper in the fifth 

 volume of the Transactions of the Royal Society of 

 Edinburgh, by Mr Wallace, of the Royal Military Col- 

 lege. 



Sect. VIII. 



On the Equations of the Conic Sections, and 

 their Identity with Lines of the Second 

 Order. 



From the property of the conic sections, which we 

 have employed in their definition, their polar equation 

 may be immediately derived. Let us suppose, as in 

 the definitions of Sections II. III. and IV. that the 

 curves are traced by D, the extremity of a variable ra- 

 Fi 3 dius FD, which revolves about F, a fixed point, as a 

 27. 6i. centre. This line has, in the application of the theory 

 of these curves to astronomy, been called the Radius 

 Vector. Put r for the line FD, z for the angle DFA, 

 which it makes with the axis; d for FP, the distance 

 of the focus from the directrix ; and let the determin- 

 ing ratio be that of 1 to n, a given number. Draw DI 

 perpendicular to the axis, and DE to the directrix ; 

 and because Rad. : Cos. z : : FD or r : FI, and 1 : n : : 

 FD orr:DE or IP, we have FI=r Cos. z, IP = wr, 

 and therefore d — r Cos. z + nr, and 

 d 



r_ M+Cos.z ; ( L ) 



This equation is common to all the sections. In the 

 Ellipse and Hyperbola, put the eccentricity CF= e ; the 

 senaitransverse axis CA=o ; then because l:n::e:a, 



ed 



(2.) 



— a -f- e Cos. z ' 

 This equation belongs to the Ellipse and Hyperbola. 



Again, because when 2=0, then r=a — e in the ellipse, 

 and = e — a in the hyperbola ; and as, in this case, 

 Cos. z = 1, wc have, in the former case, edzz a 1 — e 1 ; 

 and in the latter c d = e l — a 1 , therefore, 



In the Ellipse, rzz ~ 



a -f- e Cos.2 /g * 



M 



Hos. z t 



In the Hyperbola, r= — - — ~- 



CL ■+■ c v>OS. . 



In the Parabola, put p for the parameter of the axis., 

 then as in this case, « = 1, we have 



r ~ 2(1 + Cos.2) ( 4< ) 



for the polar equation of the parabola. 



Let us now suppose that the origin of the rectangu- 

 lar co-ordinates is at A, one of the vertices of the axis ; 

 put AI, the distance of the ordinate from the vertex, 

 = x ; ID the ordinate = y ; and in the ellipse and 

 hyperbola put BC the semiconjugate axis = b ; then 

 in the ellipse d l :b l ::x {2a — x):y\ (Sect.II. Prop.16.); 

 in the hyperbola a* : b 2 : : x (2 a + x) : y 2 , (Part III. 

 Prop. 24. ) ; therefore. 



In the Ellipse, a 2 y" 1 -j- & & — 2 a to x — 1 

 In the Hyperbola, a* y* — ¥ x* — 2 a b 2 x— > . . (5.) 

 In the Parabola, y* — p x = J 



In the ellipse and hyperbola, the centre may be 

 taken as the origin of the co-ordinates, and then put- 

 ting CI = x', so that x = a — x', we have 



In the Ellipse, a*yi + b* x' 2 — a* b 2 = 1 (( . . 



In the Hyperbola, a*y* , f —b*x'2+ a 2 6 1 =0 J " * ' *• ' 



Since it appears that the rectangular co-ordinates of 

 a conic section are in every case the variable quantities 

 of an indeterminate equation of the second degree, it 

 follows that every conic section is a line of the second 

 order : The converse is also true, namely, that every 

 line of the second order is a conic section, as we shall 

 now demonstrate. 



The equation to a line of the second order, in its 

 most general form, is 



Ay 1 +B xy + Cx* +T)y + Ex+F=0, 

 where x and y denote co-ordinates to two axes, which 

 are perpendicular to one another, (as explained in the 

 theory of Curve Lines), and A, B, C, D, E, and F, are 

 given quantities. By resolving this equation in respect 

 of y, and putting p=B 2 — 4 AC, 0=2 (BD — 2 AE), 

 r=D 2 — 4 AF, we find 



Bx + D_ J _ 1 „ 



y= — 2T~— ^x^(P xi +i x + r ) ; 



now that the equation may represent a real curve, the 

 expression under the radical must be positive to all va- 

 lues of x within certain limits. If p be positive, we 

 may suppose x to have such a value that p x 2 shall ex- 

 ceed the amount of the other terms, and every greater 

 value of x will have the same property ; so that in this 

 case, the expression may always be positive. When p 

 is negative, let the expression be put under this form, 



Then, in this case, to the values of x within certain li- 

 mits, (2px + q) x +4tpr — o l must also be negative, 



so that when multiplied by — , the product may be po- 

 sitive, and consequently, changing the signs, 

 q 1 — ipr — (Upx + o) 2 , 



