290 Sir William Rowan Hamilton on Fluctuating Functions. 



\ da cos ipa-px)f^= — LT/ ^-COS^x\ daCOS^af^; (z'") 



m being here an Integer number, which is to be supposed large, and ultimately 

 infinite. The equation (g) becomes therefore, in the present question and by 

 the present method, as well as by that of the last article, 



cos plzzO ; 



and if we make p zz p-^-y, p being a root of this equation, we may neglect y in 

 the second member of (z"'), except in the denominator 



cos §1:=. — sin pi sin 7/, 



and in the fluctuating factor of the numerator 



cos (2toj3/ -\-?l-{- ^'^) = — sin pi sin (2myl -\- yl) ; 



consequently, multiplying by tT^ dy, integrating relatively to 7 between any two 

 small limits of the forms ipe, and observing that 



lim .2^' sin(2TO/7 + /7) ^2^ 

 m = 00 7r J_, sin ly I ' 



the development 



2 



yi = r 2p cos /9^ \ da COS pa/^, 



which coincides with (y'")» ^^^ is of the form (h), is obtained. 



[16.] A more important application of the method of the last article is sug- 

 gested by the expression which Fourier has given for the arbitrary initial tem- 

 perature of a solid sphere, on the supposition that this temperature is the same for 

 all points at the same distance from the centre. Denoting the radius of the 

 sphere by I, and that of any layer or shell of it by a, while the Initial temperature 

 of the same layer is denoted by a~^J'„, we have the equations 



/o=0,/,+ ./. = 0, (a-) 



which permit us to suppose 



V being here a constant quantity not less than — /"', and/"' being the first diffe- 

 rential coefficient of the function y^ which function remains arbitrary for all values 



