in the theory of Conduction of Heat 423 



expansion is numerically equivalent to the simpler formula ob- 

 tained by the symbolical method*. 



§ 4. Evaluation of certain symbolical expressions used in 

 the foregoing sections. 



As explained already in § 1, the fundamental meanings of the 

 symbolical formulae can be obtained by translation into complex 

 integrals. In fact the meaning of the function /(r/), where 



Kq^ ^ p = djdt, 



is given by the complex integral 



where kv^ = A. 



In the first instance we are concerned only with functions of q, 

 which are in reality even functionsf; or functions which are (in 

 theory) such that/(p') is expressible as a one-valued function of A. 

 But in the symbolical transformations oif{q), it is generally con- 

 venient to manipulate algebraically, without restricting the func- 

 tions used; and then we must adopt some definite convention as 

 to the interpretations. First we make v single-valued by means of 

 a cut along the negative real axis in the A-plane; and we select 

 that value for v which has its real part positivet. 



Further the functions / (q) are (in their original forms) such as 

 to tend to zero when q ^co ; and thus the complex integral 

 above can be replaced by an integral along the path indicated 

 in the A-plane. 



When the path has been modified, we can suppose the algebraic 

 transformations of the function/ (v-) carried out so as to correspond 



* For Perry's case (w = 1) the two formulae wc^rk out as «„ (-2982) and Vq (-2816) ; 

 hut, as already remarked, the value of {Kl/a^} e-"'^"^ is then about rh^ so that a 

 discrepancy is to be anticipated unless the definite integral given is evaluated 

 numerically. 



t For instance qa coth qa can be written in the form 



Then the corresponding function of v is equal to 



2 ! K 4 ! /c^ / \ 3 ! /c o ! k^ 



X This has practically the same effect as if we regard q as real and positive, in 

 our algebraic work. 



