84; Mr. Lubbock on some Problems in Analytical Geometry. 



equations the values of «, |3, y, found by elimination, in terms 

 of A, B, C, D, E, L, M, N from equations of lines 29, 30, 31. 



If the equation to the curve surface be in the form 

 n^p^ .r' + m^p^Tf + m^ if z^ = m^ rfp^^ and a, /3, -y are the co- 

 ordinates of the point, as before, from which normals are drawn, 

 irfy {x—a) ^ rfi x (j/— |S) 

 n^z{y-^) = p^y (z-y) 

 p^ X (z—y) = nfz (x—u) 

 These are the equations to cylinders which have for their bases 

 equilateral hyperbolas, and which pass through the centre of 

 the curve surface and the point a, /S, y. 



Let x', y, z' be the coordmates of any point in the normal 

 drawn from the point a, /S, y to the point x,y,z in the curve 

 surface, 



X — a x' — a X — a a/ — a 



y-^ ~ y-^ 2-7 ~ z'-y 



Eliminating x, y, z between these equations, 



(«y-/3a:'r {«^(y-«)= + tf (y'-^f + P'(z'-r)'} 



= (n^-my (x'-«)- (y-/3)^ 

 iya^-ut/y {m^ {xf-uf + n" {jj-^f + p^ {n'-yf} 



= {m^-p^) {z!-yYix<-uy 

 (/3 x'-yy'y {m^ (x'-u)^ + «' (y-^)' + p^ (z'-y)'} 

 = (p«-mT(y-/3r(z'-y)^ 



which are the equations of conical surfaces whose intersections 

 are the normals which can be drawn from the point a, ^, y to 

 the curve surface. 



The equation to the cone which circumscribes the surface 



ffp^ 3^ + nf p^y^ + m^n^ t? = m^ n^p% 



and whose vertex coincides with the point «, /3, y, is 



n*p^{x-ay + m^p" {y-^Y + m^rf{z-yf 

 — p^{^x—eiyY—m^{ctz — yxY—Tf{yy—^zy — 0; 



and the equation to the plane of contact is 



rfp^ a.x + m~ p^ /3 j/ -f rf vf yz — m* n- p\ 



These equations relative to the circumscribing cone are given 

 in various works on analytical geometry. 

 If the ellipse 



n* x^ + m-y^ = m' n* 



be considered, and if the axes x and y, be at right angles, 



\x = m- (x—u) 

 \y=^ n'{y-^) 



and 



