Operational Methods in Mathematical Physics. 417 



Equation (32) can be satisfied identically only if r 4 = R 4 , 

 so that the condenser is shunted against the whole resistance? 

 R4 : and then the balance reduces to an arrangement given 

 by Maxwell *. Thus we find the conditions for a complete 

 balance, 



r 4 =R 4 , R 1 R 4 =R 2 R3, Li = KR 2 R 3 = KRiR4. (33) 



But, when r± is less than R 4 , w r e can still arrange for 

 zero time-integral by the conditions : 



RjR^R^, L 1 R 4 +R 1 K^(R4-n) = R 2 R 3 Kr4, 



found from the constant term and the coefficient of p in (32) ; 

 these readily reduce to the forms 



R 2 R 4 = R 2 R S , L x = KR 1 r 4 2 /R 4 (34) 



The advantages of the operational method will be seen 

 very clearly if the above simple calculations are compared 

 with those given, for instance, by A. 0. Alien f. 



Further examples of induction-balances are described in 

 Allen's paper just quoted; and any of these will be found 

 to give simple exercises in the practical application of 

 operational methods. General discussions will be found in 

 vol. ii. of Heaviside's ' Electrical Papers ' J. 



§ 3. Direct Proofs of the Equations (2), (4) 

 given in § 1 above. 



It is somewhat more natural to take Heaviside's equation 

 (4) first, in which we suppose F(p), A(p) to be simple 

 polynomials, with A(p) of higher degree than F(y?). 



Then, as a matter of elementary algebra, we can write 



— ^\ = X , where A= . > ' . . . 3d) 



A(p) * p— a ^ (a) 



* ' Electricity and Magnetism,' vol. ii. Art. 778. 



+ Phil. Mag. vol. xxv. 6th series, 1913, p. 520 ; the conditions (33), 

 (34) are considered in §§ 2, 3 of his paper. Reference may also be 

 made (for a different method) to a paper by J. P. Dalton on p. 06 of the 

 same volume. 



t See, for instance, pp. 33, 102, 260-297, etc. 



Phil. Mag. S. 6. Vol. 37. No. 220. April 1919. 2 G 



