82 Mr. Lubbock o« sojne Problems in Analytical Geometry. 



Let D— 0, E= 0, N— 0; then multiplying equations (2) 

 (3) and (4) first by x,y andz respectively, then by x—a,y—^ 

 and z—y respectively, and then by «, /3, y respectively, and 

 adding together the results, B, being the distance of the point 

 a, jS, y from the centre, and r the distance of the point in which 

 the normal cuts the curve surface, from the centre, 



_ _ i + r'-B? 



4>F 



r' = x^ + y'^+ z^-\- 2 xy cos xy + 2 zy cos zy + 2 z x cos s t 

 R^ = «2 +1324.^2 -I- 2u^cosxy + 2y^coszy + 2ya.coszx 

 X is therefore evidently independent of the direction of the 

 axes X, y, z. 

 Eliminating x, y, z from the equations 



X f2 Ax + By + Mz] = x + y cos xy + z cos x z 

 X{2Cy+Bx+ Lz} = y + xcosxy + z coszy 

 X {2Kz + Ly + Mx} = s + xcoszx + y cos zy 



which obtain upon the supposition that a, ^,y = 0, in which 



case X = - -^. 



— 2F^< — 2 BL cos zx— 2 B M cos zy — Q ML cosy x W* 



C SAKC -2 AL^-2£-B'-^CM' -2MLB 3 



{2 C sin' z X + 2 K sin 'i X y + 2 J iin'i zy 

 — 2 L (cos zy — cos zx cos xy)— 2 B (cos xy 

 — cos z X COS zy) — 2 M (cos z x — cos xy cos zy) 

 iAKC-2AU -2KB^-2CAP—2MLB 



— 82^'( 1— cos' zy — cos'iry — cos'zi+ 2 cos zx cosxy cosxy | 



~SAirC-2AL^-2KB'-2CM'- 2MLB ~ " 



If m, n, p are the principal axes of the surface, 

 the coefficient of ?'' = ?«'^ + n- + p^ 



of r^ = m^ p" + m^n^ + n^ p"^ 

 and the quantity independent of r = m^ifp^ because m^, «^ p'' 

 are evidently the roots of the equation. 



If the curve surface be referred to conjugate axes ?«', n',j>' 

 so that 



A - m"p'-, C = m'^«% K = m'^7i'^ 

 B = 0, Z = 0, M = 0, F= — m'- n'- f' 

 r" - (m" + n'^ + p'^) r* 



+ (/>'- n'^ s'm^y z + p'^ m'^ sin- zx + m'^ 7p sin ^ xy) r* 

 — m'- n'^^p)^ ( 1 — cos-zy— cos^ xy — co^^ zx 

 + 2 cos zx cos xy cos zy) ■= 0. 



Whence 



