166 



CONIC S 



of the 



Conic 

 JSectioni 



Equation* must be positive : now the last term will always be ne- 

 gative, and it may evidently become as small as we 

 please ; therefore the whole expression may be rendered 

 positive, provided that q l — 4-pr or 4 (BD — 2 AE) X — - 

 4 (B«— 4 AC) (D* — 4AF) is a positive quantity. This 

 expression when developed, and the common numerical 

 multiplier rejected, gives us 



A (AE2 + CD2+FBS— BDE— 4 AFC) 



which, in the case of P or B z — 4 AC being negative, 

 must be a positive quantity, and this condition must be 

 satisfied, otherwise the equation 



Ay-fBx#+Cx 2 -f D.y+Ex-f F=0 



cannot represent a real curve. 



Let the co-ordinates to any assumed point in the 

 curve be a and b, so that 



A62- r Ba6+Ca*+D6 + Ea_{_F=0, 



and let the origin of the co-ordinates be transferred to 

 that point, by making 



xzza+x', y—b+y; 



# and y' being new co-ordinates parallel to the former. 

 Then we have 



Ay' 2 +Bx'y' + Cx'* + (QAb + Ba + T>)y'\ 



-}-(2C«+B6 + E)x'J - U 

 ox, for greater simplicity, putting 



2Ab + Ba + D=D', 2 C a + B J + E=E', 

 Ay'* + Bx'y'-{-Cx 2 + T)'y-\-'E' x'=0. 



Let us now change the direction of the co-ordinates 

 9, y about the new origin, at the same time keeping 

 them at right angles to one another. This is done ( see 

 the article Curves) by assuming 



x'=x" Cos. * — y" Sin. «, y'z=.x" Sin. »-\-y" Cos. *, 



x" andy being new co-ordinates. By substituting these 

 values of x' andy we have 



(A Cos. 2 * — B Sin. * Cos. *+C Sin. 2 *) y"t 

 + (A Sin. 2 a+B. Sin. * Cos. * + C Cos. * *) x"* 

 , f2 ASin. *Cos. *-f-B(Cos. 2* — \ „ „ 

 "*" I Sin. 2 *) — 2 C Sin. * Cos. « J x % 

 4. (D' Cos.*— E' Sin.*)y + (D'Sin.*+E'Cos.*)x"_ 



As the angle « is arbitrary, it may be taken such, 

 that the term x" y" may disappear ; this will be the case 

 if it satisfy the equation 



2. A Sin. * Cos. *-}-B (Cos. 2 * — Sin.* *) 1 



— 2 C Sin. « Cos. * j — ' 



But we have (Arithmetic of Sines, Tables F. G.) 

 Sin. 2 *=2 Sin. * Cos. «, Cos. 2 «=Cos. 2 « — Sin. 2 * ; 



therefore the equation which determines * may be put 



under this form 



(A— C) Sin. 2 *-f-B Cos. 2 «=0, 

 which gives us 



Tan. 2*=— 



A — C 



The angle 2 * will always be real, seeing that its 

 tangent is real ; thus the foregoing transformation and 

 reduction are always possible. To introduce it into 

 the equation, it must be observed that we have in ge- 

 neral 



1-fCos. 2 * „. „ 1 — Cos** 

 2 



B 



Cos.* *=- 



Sin. 2 *=- 



Sin. 2 *= 



-/(1 + Tan. 2 2*) 



E C T I O N S. 



|a + C— B Sin.2*+(A— C) Cos. 2*1 y* 

 + JA + C + B Sin. 2 *— (A— C) Cos. 2 *1 %>'* 



+ 2(D'Cos.*— E'Sin.*) y" + 2 (D' Sin.«+.E'Cos. *) «". 

 But from the value found for Tan. *, we set 



Tan. 2 * 3 



• j>+(A-C)*]/ 



r a l A ~ C 



if the co-efficients of y" 2 and x" 1 be represented by M 

 and N, we shall find, by substituting these values of 

 Sin. 2 *, and Cos. 2 *. 



M=A + C + ^/|B l -r.(A— C) 2 |, 



n=a+c— v/Jb'-ha— cy\, 



and the equation becomes 



My 2 -f Nx" 2 -f 2 (D' Cos. *— E' Sin.*)/' 



+ 2 (D'Sin. x+E'Cos.*) x"r.*0. 

 The co-efficients M and N are always real quantities, 

 because Tan. 2 * is a real quantity ; but besides, M may 

 be taken always positive, for that this may be the case, 

 it is only necessary to dispose the terms of the equation 

 so that A may be positive, which may always be done 

 by changing the signs of the terms : And A being po- 

 sitive, if C is also positive, M will be entirely compo- 

 sed of positive quantities; but if C is negative, and equal 

 to — C, the radical part which becomes then 



Equation* 

 ot the " 

 Conic 



Sections. 



*/{b 2 +(a+c) 2 } 



and by substituting these values and that of Sin.* Cos.*, 

 we find the fallowing expression equal to 0, viz. 



is greater than the rational part, which is then A — C : 

 We shall therefore suppose M to be a positive quanti- 

 ty ; as to N, it will be positive in some cases, and ne- 

 gative in others ; and, in one case, it will be = ; for 



since N=A + C — J Jb s +(A — C) 2 } ; if (A + C) 2 



-^ B 2 -r. (A— C) 2 , that is, if 4 AC^B 2 , then N is po- 

 sitive. On the other hand, if 4 AC ^L. B 2 , then N is 

 evidently negative ; and if 4 AC=B 2 , then in this case 

 N=0; so that the equation may be expressed thus, 

 M y" 2 =±= N x" 2 + 2 (D' Cos. * — E' Sin. « ) y" 



+ 2 (D' Sin. * +E' Cos. *) x"=0. 

 As this equation yet contains the quantities D', E', 

 which are composed of the arbitrary co-ordinates a, b, 

 we may make any other assumption that is consistent 

 with the indetermination of x" and y". Let us therefore 

 assume 



2 (D' Cos. * — E' Sin. *) = ; 

 by this, y" vanishes from the equation ; so that, if to 

 abridge, we put 



2 (D' Sin.* + E' Cos. *)=P, 

 we have simply 



M/>±Ni" + P/=0 

 which evidently belongs to an ellipse if N is positive, 

 or tn an hyperbola if N be negative ; or, lastly, to a 

 parabola if N=0, the origin of the co-ordinates being 

 manifestly in each case the extremity of the axis. 



The co-ordinates a, b, at first taken as arbitrary, may 

 now be determined from the assumptions which have 

 been made ; for by multiplying the equation 2{ D' Cos. * 

 — E' Sin. *)=0 by Sin.*, it becomes 



