20 Mr. 0. Heaviside on the 



The effect of making the substitutions (8 6) in (5 6) is to ex- 

 press C in terras of the P, Q of (9 b) and the A ; B, a, b of 

 (116); thus:— 



+ { (••+ ..-..) +(..- ..-..) \e-W~4 



+ i{ (..-..+.. ) s inQ(Z-s) + (..+ .. + .. )cosQ(Z-z)}e^-*> 



+;{-(..+ .. - .. ) + (..- ..-..) \e-W-i 



X Y sin nt 

 -r [{(A+a) e ^cosQ?-(A-a)e- p/ cosQ/-(B + 6)e^sinQZ-(B-&) e - p WQZ} 

 + i\(B + b) .... -(B-6) .... + (A + a) .... + (A-a) .... }]. (1 



The dots indicate repetition of what is immediately above 

 them. Here we see the expressions for the four quantities 

 A', B', P', Q' of (65). which we require. (12 b) therefore 

 fully serves to find the phase-difference, if required. I shall 

 only develop the amplitude expression (7b). It becomes, 

 by (12 J), 



' 



(»- 



[ e 2?a-z) {(P 2 +Q 2 )+(K 2 +S 2 n 2 XR/ 2 +L/ 2 n 2 )+2Qn(It 1 'S+KL/)+2P(KR 1 '-L/S« 

 + e -2P(*-*).j .... + __ _ 



+ 2 cos 2Q(l - z) \ (P 2 + Q 2 ) - (K 2 + S 2 n 2 )(R/ 2 + L X V) } 



~-4sin2Q(Z-^){Pn(E/ + KL 1 ') + Q(L/Sn 2 -KR/)}]^ 



~ [^{(A + a ) 2 +(B + 6) 2 f +e -^ l {(A-af + (B-by\ 



-2cos2QZ.(A 2 + B 2 -a 2 -6 2 ) + 4sin2Q?.(A6-aB)}]^ 



in terms of A, B, a, b of (116). 



This referring to any point between 2=0 and I, a very im- 

 portant simplification occurs when we take z = l. It reduces 

 the numerator to 2(P 2 + Q 2 )*. It only remains to simplify the 

 denominator as far as possible, to show as explicitly as we can 

 the effect of the terminal apparatus, which is at present buried 

 away in the functions of A, B, a, b occurring in (136). 



First of all, we may show that the product of the coefficients 

 of e 2n and e~ 2P? equals half the square of the amplitude of the 

 circular part in the denominator. This is an identity, in- 

 dependent of what A ; B, a y b are. (13 6) therefore takes 

 the form 

 C =2V (P 2 + Q 2 )* -f- [G e 2P * + He- 2P '-2(GH)*cos 2(QZ + 0)]*. . (J 



