Mr. Lubbock on some Problems in Analytical Geometry. 83 



Whence 



,„'9 + „'2 -I- p'2 = 7B^ + n* + P* 



p'2 n'2 sin^y + y^ m'« sin^ z a: + w'^ n'^ sm^xy 



= p^n^ + p^ m^ + m® ?j^ 

 ot'2 n'2p'2 ( 1 — cos- z^ — cos^ xy — cos-z x + 2 cos zx cos jt?/ cos 2j/) 



= n^ n-p^ 

 which are the known properties of the conjugate diameters ; 

 the first and third of these theorems appear to have been first 

 proved by M. Livet, in the thirteenth Number of the Journal 

 de I'Ecole Polytechnique ; and the second by M. Brianchon 

 in a memoir " Sur la Theorie des Axes conjugues, et des Mo- 

 mens d'lnertie des Corps," Journal de VEcole Polytechnique, 

 vol. viii. p. 65. • i i 



If xi, y, z' are the coordinates of any pomt m the normal 

 drawn from the origin, which coincides with the centre, of the 

 curve surface, 



Ax" + C/ + Kz"- + Bxy + Lzy + Mzx + i^= 

 X y' _ X z' 



(2ila/ + 5v' + Ms') (y +^ cos A- 3/+ z'coszj/) 

 = (2Cy + 5^ + Ez') (jt' + y cos xy + z'cos JTz) 



(2 CiJ + Bx' + L:^) {z + a:'coszj7 + y cos zy) 

 = {2Kd + Ly' + Mx') {y' + x' cos xy + x^ cos zy) 

 which are equations to conical surfaces whose intersections are 

 the principal axes of the curve surface, 



Ax^' + Cf + Kz' + Bxy + Lzy ■\- Mxz + F - 0. 



Transferring the origin to the point a, /3, y 



Ax"- + Cy2 -^K:? + Bxy + Lzy + Mxz + Z)a- + £3/ + 



ATz + jp=0 



Z) = -2yla-i?/3 — Cy 



£ = -2C(3- J5a - I/y 



^■= -2Ky- L^ - Mcc 



{2Ar'-^B!/ + Mz' + n} {y-/3+(x'-«)cosxy+ (='-y)cos.y } 

 = {2Cy+i?x' +X«'+£} {x'-. + (y-/3) cos rry +(»'->) cos x«} 



{2 Cy + 2?x' + I- «' + E} {z- y + («'— ) (cos»:« + (y'"^) ^""^y} 

 = {2A'.'+/.y +A/x' + iV} |y_/3 + (x'-«) cosxy + (='-y)cos.yl 



In order to have the equations to these conical surfaces 

 which determine the principal axes in the most general case 

 possible, it is only necessary to substitute m the precedmg 

 * ' •' 1^ 2 equations 



