in the theory of Conduction of Heat 427 



When/ (5') is converted into a complex integral along the path 

 indicated above, there is a contribution from the small circle round 

 the origin, which tends to the value Aq, when the radius of the 

 circle tends to zero; and the straight paths contribute the integral 



Where ^ (m = /(-^) "/(- -^) ^ ^« 



wnere f^u) ^.^ (1 _ 5)2 + 52^2^2 • 



Thus f{q) = AQ+ - \ </. (6) e-^'^'f dd 



gives the interpretation of the function/ ((7). 



Now (/> (d) can be expanded in powers of 6, when 6 is sufficiently 

 small; and, having regard to the connexion between ^ (d) and 

 / (5-), it is clear that the expansion will be of the form 



cf>{d) = A,-A,d^ + A,d^-.... 



Denote the sum of the first n terms of this expansion by 

 Sn (d) ; then it is easy to see that we have the algebraic identity* 



Thus for all real values of 6, the difference between 6 (6) and 

 Sn{d) is numerically less than (— 1)'' A^n+id^""; and accordingly 

 the interpretation oif{q) may be taken as 



9 r ^ 

 A + - Sn (9) e-^''-'dd, 



with an error which is numerically less than 



- H- irA,,,^,9^^e-^''Uld. 



TT J Q 



Thus we may say that the error in replacing/ (5-) by 



A, + A^q + A^q^ + ... + ^2„_i ?2'*-i 



(or by Aq + A^q + A^q^ + ... + A.^^q^^) is less than the numerical 

 value of the following term Aon+i 9^""^^ (when interpreted according 

 to the rules already given). This establishes the asymptotic 

 property assumed in regard to the series ioTf{q). 



* On multijilying up, we see that 



is a polynomial of degree 2n in d; further when is small it can be expanded in 

 powers of 0, the first term being ( - I)" (1 -5)^ ^2,1+1 0-". Accordingly the product 

 must reduce to this single term. 



