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XXXV. Examples of Operational Methods in Mathematical 

 Physics. By T. J. I'a. Bromwich, Sc.D., F.R.S.* 



THE use of operational methods has heen repeatedly 

 urged, in electrical and other physical problems, by 

 Heavisidef. It does not, however, appear that the method 

 has been as widely used as it deserves to be, both on account 

 of its simplicity and its elegance ; and even when the method 

 has been used, some hesitation has been shown in accepting 

 the conclusions based upon this process. 



Attacking certain problems by a different method, I was 

 led to some general rules which have confirmed the accuracy 

 of Heaviside's method J ; and I have recently developed a 

 more elementary treatment of the method (given in § 3 

 below). 



In order to illustrate the merits of the method a number 

 of examples have been worked out in Conduction of Heat and 

 in Electricity. Some of these will be found in §§1,2 below: 

 those in § 1 are suggested by the problem of thermometers 

 carried by aeroplanes. It will be found, I think, that these 

 solutions are more direct, as well as being shorter than those 

 which have been given previously ; it is easy, moreover, to 

 take account of the effect of surface-conductivity without 

 adding substantially to the difficulties of the discussion — 

 see, for instance, examples (ft), (7) in § 1. 



§ 1. Some Operational Solutions in Heat-problems. 



In connexion with recent observations on thermometers 

 carried on aeroplanes, it has been suggested that the surface- 

 conditions may be represented by supposing the temperature 

 in the surrounding atmosphere to be proportional to the 

 time t. This represents the assumption of a uniform rate 

 of descent and a uniform temperature - gradient in the 

 atmosphere; and solutions of certain problems under this 

 hypothesis have been worked out by Mr. A. R. McLeod § . 



It is easy to apply operational methods to solve such 

 problems, using an interpretation of the formulae which is an 

 extension of a known formula due to Heaviside : fairly 

 simple direct proofs will be given below (§3) both for the 

 new formula and for Heaviside's original result. But in 



* Communicated by the Author. 



f Electrical Papers and Electromagnetic Theory, passim. 

 X Proc. Lond. Math. Soc. vol. xv. ser. 2, p. 401 (1916). See in 

 particular, §§3, 4. 



§ Phil. Mag. vol. xxxvii. ser. 6, p. 134 (January 1919). 



