THE 



PHILOSOPHICAL MAGAZINE 



AND 



ANNALS OF PHILOSOPHY. 



[NEW SERIES.] 



AUGUST 1831, 



VII. On some P^-oblems in Analytical Geometry. By 

 J. W. Lubbock, Esq. V.P. S^ Treas. R.S.* 



LET the equation to any curve surface of the second order 

 be 



Ax'^ + Cy^+Kz^ + Bxy + Lzy + Mxz + Dx + Ey+Nz + F=0, 



{F(x,y,z) = 0) 

 the coordinate axes a:, ?/, z being inclined to each other at any 

 angle, and let g be the distance of any point (a, /3, y) from 

 {x,yf z), {x,y, z) being situated on the curve surface. 



q^ = (x-^r + o-iS)^ + {z-yr 



+ 2(x— a) (j/— /3) cosxy + 2 (j/— /3) (z—y) cos zy + 



2(z— y) (x—a) COSJTS (1) 



If g is a maximum or minimum and X is some indeterminate 

 quantity, 



>,{2Jx-\-By+ Mz+ B} = x—a + (3/— /3) cosxy -t- (: — y) costs (2) 

 x{2Cy+JBx+ Lz + E} =y— /3 + (a?— «) cosxy + (z-y) cos«y (3) 

 X {2Kz + Ly + Mi + iV^} = z-y+ {x—«)coazx + (y-/3) coszy (4) 



Eliminating A, 



{2A X + Sy + Mz -t-D}{y-|3i- (r-a) cos ly + (s — y) coszy} 

 = {2Cy + J?i+ Lk+ E\{ x-x + {y-li) cosxy -\- (z-y) cosxz] (•'') 



{2Cy + i?i + iz + £} { z—y + (i— a) coszi+ (y— /3) cosKy) 

 -{2A'z + Xy + Af I -t- iV} { y-/3 -f- (!-«) cosxy + (2-7) cos zy} (6) 



which are the equations to curve surfaces wliose intersections 

 determine the points from which the normals can be drawn 

 from the point «, /3, y to the curve F{x,y, z). 



* Coininunicatcd by the Author. 

 N.S. Vol.10. No. 56. //?/i,^ IH'Jl. M Let 



