in the theory of Conduction of Heat 425 



and by a known result, due to Cauchy, 



2mj„_,^Ai+'" r(l + m) 

 if 1 + m > and t > 0. 



Hence 1 _ VH _ Vl'c^J _ 2 /M 



^" 'z ~ r (t) - ir (1) - V U 



a result which follows symbolically from (i) by observing that 

 Ik i dt 2 



(vi) Interprelation of l/q'^. 



The method of (v) shows at once that 



1 _ {Kt)'^+'^ 1 _ (/cO"* 



and we can sum up all the preceding formulae in the single state- 

 ment n 



^ _ {Kt)2 



q^~ r(i +^)' 

 which may be stated even more simply in the shape 



1 _ P 

 ^~ r (1 + w)' 



where n may be taken as having any (positive or negative) value. 

 The last result is obvious by successive integration when 7i is an 

 integer: and it is natural to conjecture that the same formula will 

 be vaHd generally, 



(vii) Interpretation of qe~^^, ivhere x is positive. 

 This is readily obtained by expansion in powers of x; and the 

 result is ^2 ^3 ^4 



q-q^x + q^ -, - q^ ^, + ?' 4] ~ •'•' 



Using the results found above we obtain the series 

 1 (. _ J^ «! 1 . 3 X* 1 . 3 . 5 a;« ) 



ViiTKt) { 2/cf 2 ! ' {2Kt)^ 4 ! {2Ktf 6 ! ' 



ViTTKt) I 4:Kt^2\ \4:Kt) ' 3 ! U/C^J ,1 Vi^^t) 



If these operations appear to need further justification, it is easy 

 to see that the direct expression of gc^'^* by means of a complex 

 integral leads to the formula 



^ re-"*'"- cos (dx) de = ,.^-, e-^V4«t. 



TT J y{7TKt) 



