392 Messrs. K. Honda ami S. Shimizu on the Cliange of 



Comparing with (25) T\e see that the work done by J' on 

 the motion due to an equal J is the same as that done by 5 

 itself^ as should evidently be. 



I£ in (29) ^ = 90°, so that the forces are perpendicular to 



the line joining the two ])oints of application, we get 



As we have seen, when hr is small (31) is finite and positive. 

 It vanishes when 



^^^^^^■^■=r5v- ^^-^ 



and this occurs first in the second quadrant. 



In general, when there are a number of forces acting at 

 detached points, the whole work done must be obtained by a 

 double summation of {2^) . If the forces are continuously 

 distributed, the sum becomes a double integral. A particular 

 case, of interest in connexion with the problem of electrical 

 vibrations along a circular wire^. occurs when the forces act 

 tangentially at the various points of a circular arc. Here 

 6' = 6, and {2S) becomes 



Jr^%' r 1 -f cos- e /sin kr \ sin Jrr . i 



STfbp 

 Aug. 2, 



r 1 -f cos- a' /sm A-r , \ sm A/' . ..^ ] 



LI. Change of Lenc/th of Ferromagnetic Suhstances under 

 High and Low Temperatures hy Magnetization. By K. 

 HoxDA, Higakuhakushi, and S. Shimizu, Rigahushi]. 



[Plate XV.] 



THE interesting experiments of Hopkinson on the mag- 

 netization at high temperatures promise us some 

 important results in the magnetic change of length at such 

 temperatures. On account, however, of the experimental 

 difficulties, this interesting subject has as yet scarcely been 

 investigated. Barrett! was the first to touch the subject : 

 with a rise of temperature of about 50° C. he observed no 

 effect on the magnetic change of length in iron and cobalt, 



* Compare Pocklinorton, Proc. Carnb. Phil. Soc. ix. p. -324 (1897) : 

 'Nature,' Ixii. p. 486 (1903;. It would seem that (33) must lead to a 

 more complicated expression for the energy radiated than that in 

 Dr. Pocklington's investigation. 



t Communicated by the _Authors. 



X Barrett. Phil. Alas:. "4^ xlvii. p. ol (1874) : 'Nature.' xxvi. pp. olo 

 586 a^^'2) ; Beibl vii.>."201. 



