338 NOTES ON TRILINEARS. 



which we will write for the moment, 



■2" 



up + vm^ + VM^ = . 



The semi-axes being the greatest and least values of r, we must 

 make r a maximum or minimum by the variation of I, m, n, subject 

 to the condition 

 sin 2 -4. Z» + sin 2 B. m^ + sin 2 C. w^ = 2 sin A sin JB sin C 

 a I + h m + e n =0, 



Hence we obtain 



uldl + anal + . . . . =0, 

 sin 2 A. Idl + .. .. + . . . . =0, 



adl + 4- = ; 



multiplying the two latter equations by arbitraries, X, /x, adding, and 



then equating to zero the coefficients of the differentials, we obtain 



ul + X sin 2 A. I + ix a =■ 



vm + X sin 2 B. m + fi h = 



ton + X sin 2 C. n + fj. e = 



Multiplying these respectively by I, m, n, and adding, we obtain 



1- A- 2 sin ^ sin J5 sin C = 0, 



or, 



X = 1 . E. 



2 sin A sin B sin C r^ ' 

 also, transposing these equations, 





ft a '^ sin 2 ^ 



H- 



u -\- X sin 2 A « , ^ 



sin 2 ^ 



b 



sin 2 B 



sin 2 B 



+ X 



r- n 



sin 2 C 



+ X 



sin 2 G 

 Multiplying these respectively by o, b, c, and adding, we obtain 



_ ^ anal + . 



