370 REPORT — 1893. 



If we take for an instant the plane of (xy) to be the interface, so that 

 (Z, m,n)^(0, 0, 1), this expression becomes 



Now on any form of interpretation of MacCullagh's theory, no ex- 

 traneous interfacial forces at all are required to satisfy the boundary 

 conditions ; if the present theory is to agree with it, we might expect that 

 there will be required only interfacial forces such that their activity will 

 for the actual motion just undo the variations of this surface-energy. 

 But the boundary conditions of MacCullagh are (§9) 



t, »?, 4, -5- and --- 

 4g ah 



all continuous, where UcZr is the statical energy ; and these do not 



suffice to make this surface- energy constant, i.e., the time variations of 

 the above expression continuous across the interface. As already re- 

 marked, the theories of Kirchhoff and MacCullagh are formally identical ; 

 therefore there must be some discrepancy here. It is in fact the circum- 

 stance that this surface-integral part of the energy has lost its correct 

 location, and does not really belong to the place with which it is now 

 analytically associated. 



Again, KirchhofiF's actual procedure is to take the tractions (X, T, Z) 

 and (X', Y', Z') on the two sides of the interface that are derived in 

 Lagrange's manner from the energy-function, and to equate to nothing 

 their activity 



(X-X')^-l- (Y-Y')'^^+(Z-Z')'?. 

 at at at 



If ^, rj, Z are quite independent this will give three boundary conditions 

 just as before, and will be no help. Bat in the motion to which he 

 restricts himself, I, i], ^ are the displacements in a plane-wave, and so are 

 functions of the same linear function of x, y, z and t ; he finds that the 

 introduction of this restriction reduces the conditions to two, and so 

 allows further progress. 



The reason which Kirchhoff assigns for the two theories of himself 

 and MacCullagh being analytically in agreement is that they can only 

 differ as to boundary conditions, that he gets to a definite theory by his 

 principle of extraneous forces, and that MacCullagh's definite theory also 

 satisfies this principle from the simple fact that there are no extraneous 

 forces. But then the energy-functions are not the same in the two 

 theories. The Fresnel laws of reflexion are obtained by JSTeumann really 

 by the hypothesis that for rays, i.e.. for simple wave-trains, no loss of 

 energy occurs in the reflexion. This is a much narrower principle than 

 its generalisation by Kirchhoff ; and, as we have seen, to make his 

 generalisation work, the latter has to return practically to Neumann's 

 form in which it is restricted to plane- waves. 



These considerations are set forth as showing the artificial character 

 of Kirchhoff's principle, and illustrating the various mistakes and mis- 

 conceptions which may arise in connexion with a subtle point of analytical 

 dynamics, of which the physical bearing has not, I think, been realised 

 by many of the writers on this subject. 



