(1) 



314 Mr. W. P. Boynton on the 



Assuming that the solutions are of the form 

 Qi = e K \ Q 2 = ke xt , 

 our equations become 



1 + L^X 2 + kMKjs? + R^X = 0, 

 H^L 2 K 2 \ 2 + MK 2 X 2 + e 2 K 2 X = 0. 



Solving for k, 



1 + L 1 K 1 \ 2 + R 1 K 1 \ _ MK 2 X 2 



~ MKiX 2 ~ 1 + L 3 K 2 X 2 + R 2 K 2 X' * 



Clearing of fractions, collecting, and dividing by the coeffi- 

 cient of X 4 , 



a 4 i >3 ^i^2 + L 2 Ri ^ L^ + LgKg + RiRaKxK., 

 + LiLa-M 2 +A KjK^L.-M 2 ) 



RA + R.K, 1 _ 



which is of the fourth degree in X. We are interested in the 

 imaginary roots, which occur, if at all, in conjugate pairs, 

 since the coefficients of our equation are all real. If we 

 write these 



X x = —a + i@, X 3 = — y + i8, 



X 2 = —a — i{3, X 4 = —7—^8, 



then by the theory of equations 



_(X 1+ A 2 +X 3 +^)=2«+2 7 =y|±M? = A, (2) 



SX r X g = XiX 2 + X 3 X 4 -f- (Xj + X 2 ) (X 3 + X 4 ) 

 (r^s) = a 2 + £2 + y> + gs + 4a7 



_ Lx^ + LaKa + RAKA __ 



— SX r X 3 X^ = — (Xj + X 2 ) X 3 X 4 — (X 3 + X 4 ) XiX 2 



irju&t) =2 a ( 7 2 + g 2 )+27(a 2 + /3 2 ) 



BaKt + BnK, 



(3) 



~ KiKgCI^Ljj-M*) 



\tW,-(^+/8»)(7»+8 i ) 



1 



-0, (4) 



= » (5) 



