191 1.] LINEAR DIFFERENTIAL EQUATIONS. 285 



by integration, from the solution IV (i) above. All his results 

 occur directly in the above tables or are combinations of two of these 

 solutions. It is evident that there are several misprints in the results 

 as printed in the " Britannica." 



Of course there are many solutions of Fourier's equation which 

 must be built up from elementary solutions, however found, by 

 means of Fourier series, or which must be obtained by the methods 

 of harmonic analysis. 



The solution III (5) above is the series used by Sir William 

 Thomson in his solution of the problem of the secular cooling of the 

 earth.^ 



An interesting result in pure mathematics is obtained as follows : 

 Sir William Thomson shows that for a continued point source of 

 heat, if the rate is an arbitrary function of the time, f(t), the solu- 

 tion of Fourier's equation when K = i is given by the definite 

 integral 



V 



= CdxAt-x) 



Jo 



bTT'-X"- 



The second part of the general solution (B) above shows that 



is also the solution of Fourier's equation for the same conditions. 

 It follows that 



/»« fix I r I r f^ "I 



i ''-/(' - -) 8^1 = 4^ [7 /W +^'« TT. + /"(') 4 !- + • ■ ■ J 

 is a general formula for computing the definite integral. 



Lehigh University, 

 Bethlehem, Pa. 



* " Mathematical and Physical Papers," Vol. III. 



