38 Mr. L. C. Jackson on the Theory of the 



This is a complete equation of" the sixth degree, of which r 

 for our present purpose, we can assume a solution in the 

 form 



p = iqztco i or irztco 2 or isztco , 



in which q, r, and s are small quantities whose squares are 

 negligible. 



We thus obtain the equivalent equation 



(p.-iq— ^(p-iq + ay^p — ir— & 2 }(p — ir + to^p-is— o> s )(p — is+ 0*3) = ® 



or, on expanding and omitting the negligible quantities, 



p* - 2pH [q + r + s) - p 4 (o>i 2 4- co 2 2 + o> 3 2 ) 

 + 22J s i{a) 1 2 {r + s) + a> 2 2 (s + q) + CD 3 %q + r)} 



+p (&>i 2 ft) 2 2 + &>2 2 G>3 2 + ft)3 2 W! 2 ) 



— 2pi(a) 1 2 co 2 2 r + (02 2 (Os 2 q-\-(o 2 2 (Oi 2 s) — (o l 2 o) 2 2 cos 2 = 0. (6) 

 Comparing coefficients in (5) and (6), we obtain 



1 = 2M 12 M 23 M 31 - LjLsLs + L^fe 2 + L 2 M 31 2 + L 3 M 12 2 , . . (i.) 



2(q + r + s) 

 = njj 2 h 3 ^L l R 2 Ls + L 1 L 2 Rs-R l MoJ-R 2 M. dl 2 -^ 3 M l2 2 , (ii.) 



■(co 1 2 + co 2 2 + co z 2 ) 



LJj.-Mu 2 , L 2 L 3 -M 23 2 L^-M^ 2 



+ 



+ 



+ L^R^R.^ 



0, ' -0, ' C 2 ■"1"«^ 



+ R 1 L 2 R 3 + RxRaLs, 



-2{a l 2 (r + s) +G) 2 2 (s±q) +eo 3 2 (q + r)} 



_ R 1 L 2 J-L 1 R 2 R 2 L 3 + L 2 R 3 B 3 L 1 + L 3 R. 1 



3 + C x 

 — (g)i 2 <» 2 2 + &) 2 2 a)3 2 + &) 3 2 «i 2 ) 



-f 



( 1 



V- 5 



_ L^ L 2 L 3 RiR2 R 2 R 3 R3R : 



U 2 C 3 O1O3 (-^C^ (>3 V/i C 2 



2(o) 1 2 ft) 2 2 5 + w 2 wz 2 q -t co^co^r) 

 C2O3 0x03 C 1 C 2 ' 



— G>i 2 C02 2 <03 2 



C 1 2 C 3 



(iii.) 



+ R,R 3 R3, . (hO 



>• (7) 



(v.) 



(vi.) 

 (vii.)J 



