ON THE AXES OP A CONIC IN TRILINEARS. 389 



The products, win can be eliminated from this expression by 



aid of condition (1) ; thus 



aZ + 6ot = —en 

 and therefore 



2allm = c^n^ - a^P - i^m^. 

 Hence, making 



a 



H —u -\ (au — bv — cw), 



he 

 h 



K =.v •\ (bv — cw —mi), 



ca 



c 



Z = w-\ {cw' — au — Iv)) 



ah 

 the aboTe expression becomes 



P 



— = ^Z2 + Km^ + Ln^ (3) 



To determine the axes, r is to be made a maximum or minimum 

 by the variation of I, m, n, subject to the relations (1), (2) ; 



hence, 



= Hldl + Kmdm + Lndn 

 = adl + bdm + cdn 



= sin 2 A. Idl + sin 2 J5. mdm ■\- sin 2 C. ndn, 

 and using indeterminate multipliers \, fi, we obtain 



SI + Xa + ixsm2 A.l = (4) 



Km + Xi + [Ji sin 2 B.m = (5) 



Ln + Xc + [Ji sin 2 C.n = (6) 



Then, Zx (4) + »ix (5) + nx (6) gives 

 P 

 1- /A 2 sin ^ sin J5 sin C = 0. 



Substituting this value of fi in (4), (5), (6), they become 

 1 = =1^^^ 



sin 2 ^ . P 



2sin J sin £ sin C r^ 

 m = anal 

 n = 



