2'.)X Lord Rayleigh on the' 



1*14 seconds: the value of 7 obtained in this series was 

 1*291, showing- that the thermometer had heated up con- 

 siderably after 1*14 seconds. 



I desire to express my best thanks to Prof. Callendar for 

 his advice and encouragement throughout the course of the 

 work ; also to Prof. Porter I am indebted for many valuable 

 suggestions. 



XXI. On the Spectrin}/ of an Irregular Disturbance. 

 By Lord Rayleigh. O.M., F.R.S* 



IX my paper " On the Character of the Complete Radiation 

 at a given Temperature"!, I have traced the conse- 

 quences of supposing white light to consist of a random 

 aggregation of impulses of certain specified types, and have 

 shown how to calculate the distribution of energy in the 

 resulting spectrum. The argument applies, of course, to all 

 vibrations capable of propagation along a line, and it is con- 

 venient to fix the ideas upon the transverse vibrations of a 

 stretched string. Suppose that this is initially at rest in its 

 equilibrium position and that velocities represented by <£(<(') 

 are communicated to the various parts. The whole energy is 



proportional 



f + ""W*)}. s 



to I '\<b(x)Y 2 dx: and it is desired to know 



how this energy is distributed among the various components 

 into which the disturbance may be analysed. By Fourier's 

 theorem. 



7T 0O) = j /i (jfc) cos kx dk + I f, (k) sin kx elk. . (1) 

 J Jo--.. 



where 



/;(/,) = i + %os kv 00) dv, f 2 (k) = f + °° sin kv 4>(v)dv. (?) 



It was shown that the desired information is contained in 

 the formula 



r + "{*wp^=ir"[{/ 1 w} 2 +{/ 2 (A)} 2 ]^. . (3; 



J —oc %) 



As an example, we may take an impulse localized in the 

 neighbourhood of a point, and represented by 



*(*)='"*' (4) 



* Communicated l>v the Author. 



t Phil. Mag. xxvii. p. 460 (1889) ; Scientific Papers, iii. p. 2Q8. 



