320 Mr. W. P. Boynton on the 



where E and F may be complex, and 



\=-cc + /3i, ii=-a—{3i, 



where a and $ are real and greater than 0. 

 Then 



EV 2 ^ FV^ 2EF<^ + ^ 



2\ 2fi X + fju 



E V 2 *' -f F 2 \e 2 ^ 2EF^+*>« 



2X/4 A, + fJU 



where all the denominators are real, or in terms of a and /3 



e-2«'{[E 2 (-af/3i) + F 2 (-a-y80]cos2^ 



= + ^[E 3 (-a + /3{)-F 2 (-q-ffl)] sin 2^} 



2(* 2 +/3 2 ) 

 2EFg~ 2 ^ 

 + -2« ' 



Since the oscillation is real, 

 y = e-^[(E+F)cos^ + i(E-F)sin^] = e- a< (Acos^4-Bsin)80; 



substituting A and B from this identity, the imaginary parts 

 vanish, and 



e-^\{ (B 2 -A> + 2AB/3} cos 2 fit 



fV2 M +{(B 2 -Ag)/3-2AB4sin2fr] 



J V m ~ 4(a 2 + /3 2 ) 





(A 2 + B 2 > 



4* 



If, now, j3 is large in comparison with a, the first term may 

 be disregarded in comparison with the last, and in particular 



A 2 4-TC 2 



MHt= A . + J3 (24) 



i 



(b) Of two superposed oscillations each gives in the integral 

 terms of the form deduced above ; but the terms arising from 

 the cross products of terms with different periods and decre- 

 ments require especial investigation. Such a typical term is 



J fA+V 



