Conic Sections. 



259 



of the conic section ARH relative to the axis of abscissas AM, be 

 represented by (t), (u), and QK,HKj the rectangular coordinates 

 of the same point H relative to the axis of abscissee QK and origin 

 Q, by (f) (u'). Draw RF parallel to AM, and put Z.VRF=w, 

 ZKCM=(p, Ah=d, AG=yl^ RQ=^, QG=^, RV=x, and VT, 

 the common section of the planes HTZ and RVT, equal to (y). 

 Since ARH is a conic section, the relation between {t) and (u) may 



be exhibited by the equation u^=—^ {pt-{-ct^ ;) or a^u^ —pb^t — 



cb-t^ =0 ; which evidently characterizes a right line when p=0 and 

 c=l ; a circle when a—b=p and c= — 1 ; an ellipse when (a) and 

 (b) are unequal, a=p and c= — 1; an hyperbola when a~p and 

 c=l ; and a parabola when a—p and c=0. Draw KE parallel to 

 HM and QE,KI perpendicular to KE, meeting KE and HM re- 

 spectively at E and I. The relation between (t^) (W) may be de- 

 termined, agreeably with the usual method of transforming coordi- 

 nates, by putting ^=AM=AG+QE-lK=^H-cos. cp.^' — sin. cp.u', 

 and w=HM^QG+KE+HI=^ + sin. cp.f +cos. (p.u', and substi- 

 tuting these last values of {t), (u) in the above equation of ARH. 

 From this substitution we have 



aH^ -52 (^pJc-j-cTi^) ') f h ^ 



+ [2a-H sin. cp-b^ {p-\-2ck) cos. (p]f | 

 + [2a2^cos. 9 + ^2 (^p-\-2c'k)sin.cp]u' I 



< 



+ 2(a^ 4-<^^") sin. 9 cos. cs^t'u 

 + [(a2+c52) sin.(p2_c62j^/2 



-f- [ — (a^ Arcb^^ sin. 9^+0^] W- 



where h, S, e,f, m, w,, represent the corresponding coefficients in the 



left member. Resplving the abridged equation (1) in reference to 



h 

 -]-§f 

 -{-eu' 

 ^ft'u' 

 + mf^ 

 -\-nu'^ 



Ho--(i); 



e-\-ft' 



2n 



d= 



2n 1 



h + St'-\-mf 



But 



(m') we obtain 1'/ = 



(drawing RP parallel to CK meeting HK at P) ZVRP = w-(p, 

 and consequently ^^ = QK=RP=cos. (w - 9). ^ ; therefore, supply- 

 ing the place of (f) by cos. ('^j — 9)-X in the equations above, there 

 e+/cos. (w-9).x . /fe+Zcos. (w-9)x^^2 



results u'-- 



2n 



2w 



7t + (5 COS. (w — 9)x4-w^cos. {oj—cp)-x^\ 



(2). The plane HTZ, 



being perpendicular to CK, is perpendicular to the plane ARHM, 

 and is also a circular section of the solid. The planes HTZ and 



