210 Lord Rayleigh on a Theorem 



magnetic stress B 2 /87r, is there spoken of as if it were of the 

 same nature as a simple longitudinal stress of compression, 

 producing a contraction of the length in consequence of the 

 elasticity of the metal.. . .But I see no ground for treating this 

 purely hypothetical strain as a ' correction ' to be applied, 

 either one way or the other, to the observed changes of 

 length." Mr. L. R. Wilberfbrce concurred in this opinion. 

 To answer these criticisms it was necessary to devise an 

 experiment that would isolate this effect from all others. As 

 I saw no way of doing this in the case of iron I turned to the 

 analogous case of a dielectric in an electrostatic field ; for 

 since Quincke had stated that the variation in the length was 

 proportional to H 2 /87r, it seemed that most of the effect was 

 due to this stress. My recent experiments compel me to 

 think that my original correction was erroneous and that 

 Professor Ewing was right. 



XV. On a Theorem analogous to the Virial Theorem. 

 By Lord Rayleigh, F.R.S.* 



AS an example of the generality of the theorem of 

 Clausius, Maxwell f mentions that "in any framed 

 structure consisting of struts and ties, the sum of the pro- 

 ducts of the pressure in each strut into its length, exceeds 

 the sum of the pi'oducts of the tension of each tie into its 

 length, by the product of the weight of the whole structure 

 into the height of its centre of gravity above the foundations." 

 It will be convenient to sketch first the proof of the purely 

 statical theorem of which the above is an example, and 

 afterwards of the corresponding statical applications of the 

 analogue. The proof of the general dynamical theorem will 

 then easily follow. 



If X, Y, Z denote the components, parallel to the axes, of 

 the various forces which act upon a particle at the point 

 .#, y, z, then since the system is in equilibrium, 



2X=0, 2Y=0, 2Z = 0. 



If we multiply these equations by se, y, z respectively, and 

 afterwards effect a summation over all the particles of the 

 system, we obtain a result which may be written 



2[.v .%X+y .$Y + z .2Z]=0. . . . (1) 



The utility of the equation depends upon an alteration in 

 the manner of summation, and in particular upon a separation 



* Communicated by the Author. 



f ' Nature,' vol. x.'p. 477, 1874; ' Scientific Papers,' vol. ii. p. 410. 



