Sir William Rowan Hamilton on Fluctuating Functions. 293 



the sum extending only to those roots of the equation (e^'') which are greater 

 than 0. In the particular case (m^''), in which the root (n^^) of the equation 

 (e^'') must be employed, the term (o^'') becomes, by {c'^) and (d^^), 



.||{^°^Wa«c-/(/„^.+/„_Oac]-, (q-) 



in which, at the limit here considered, 



but also, by the equations (b^''), (m^*^), 



the sought term ofy^ becomes, therefore, in the present case, 



and the corresponding term in the expression of the temperature x'^fx is equal 

 to the mean initial temperature of the sphere ; a result which has been otherwise 

 obtained by Poisson, for the case of no exterior radiation, and which might have 

 been anticipated from physical considerations. The supposition 



»'^+l<0, ' (u^'') 



which is inconsistent with the physical conditions of the question, and in which 

 Fourier's development (p^O may fail, is excluded in the foregoing analysis. 



[17.] When a converging series of the form (h) is arrived at, in which the 

 coefficients of the arbitrary function f, under the sign of integration, do not 

 tend to vanish as they correspond to larger and larger roots p of the equation (g) ; 

 then those coefficients 0^„,p must in general tend to become fluctuating functions 

 of a, as /9 becomes larger and larger. And the sum of those coefficients, which 

 may be thus denoted, 



2p0x.a,p=^^.a,p> (l) 



and which is here supposed to be extended to all real and positive roots of the 

 equation (g), as far as some given root p, must tend to become a fluctuating func- 



