292 Sir William Rowan Hamilton on Fluctuating Functions. 



,»+' 



(i-) 



Rp = (1/ COS pi — p sin pi) \ da sin paf^ 



+ sin /»«(/„+, -{-/_,). 



The equations (b^^) show that the quantity r^ does not vary witli a, and there- 

 fore that it may be rigorously thus expressed : 



Rp = 2 (1/ cos pi — p sin pl)\ da sin paf^ ; (t^' ) 



we have also, by (e^''), p being > 0, 



2(1/ COS/)/ — pmip l) 2/> ■ .jy 



cos pl-\-l [v COS /)/ — /9 sin pi) pi — sin pi cos pi' 



And if we set aside the particular case where 



the term corresponding to the root 



P=0, (n-) 



of the equation (e^''), vanishes in the development ofy^^ ; because this term is, 



by {gn, 



''-^d^{p^^'^da^m^af}j, ' (0^0 



a being very large, and j3 small, but both being positive ; and unless the condi- 

 tion (m^'') be satisfied, the equation (c^^) shows that the quantity to be integrated 

 in (0^''), with respect to p, is a finite and fluctuating function of that variable, so 

 that its integral vanishes, at the limit a =1 00 . Setting aside then the case (m'^'^^), 

 which corresponds physically to the absence of exterior radiation, we see that the 

 function y^, which represents the initial temperature of any layer of the sphere 

 multiplied by the distance x of that layer from the centre, and which is arbitrary 

 between the limits a: = 0, a: =^ l, that is, between the centre and the surface, 

 (though it is obliged to satisfy at those limits the conditions (a^^) ), may be deve- 

 loped in the following series, which was discovered by Fourier, and is of the 

 form (h) : 



2p sin px \ da sin paj"^ 

 '' pi — sin/)/ cos/)/ ' 



