!':■: 



16 Dr. C. E. Weatherburn on Two Fundamental 



a kernel becoming infinite of the second order, to discard 

 the natural idea of surface traction for that of " pseudo- 

 tension," a concept which has no physical significance. 

 This is equivalent to eliminating the anti-self-conjugate part 

 of the double stratum dyadic. I hope to show that this 

 kernel does not become infinite of too high an order to be 

 treated by the methods of my paper already referred to. 



The problems may be attacked from an entirely different 

 point of view, by the use of the Green's functions for 

 Laplace's equation. According to this method * the cubical 

 dilation 6 and the molecular rotation R are regarded for 

 the moment as known, and the displacement D is found 

 in terms of one or both of them by the aid of the Green's 

 functions for the region occupied by the body. Then, 

 taking the divergence and the curl of the value of D so 

 found, we deduce, in the case of the first problem, a single 

 integral equation for 6 from the solution of which the value 

 of D is deduced. The treatment of the second problem gives 

 a pair of integral equations, one scalar and one vector, from 

 the former of which 6 is obtained in terms of R ; then this 

 value of 6 substituted in the second gives a single vector 

 integral equation for determining R. 



§ 2. The Equations of Equilibrium. — Confining our atten- 

 tion to isotropic bodies we shall sustain no loss of generality 

 by assuming the bodily forces zero; for it is well known 

 that the equations of equilibrium can always be reduced to 

 the form in which the terms representing these forces do not 

 appear f. The ordinary equations of equilibrium are 



2 . 7 ^^ A 

 v O^ 



o// 



while the conditions to be satisfied at the surfaces are ex- 

 pressed by 



_ t _ du } g dx / ~dv dy ___ dt? dx \ fow dz _ "die dx \ 

 dn dn \~dx dn ~dy dn) \ftx dn ~&z dny 



* Cf. Boggio, Atti Lincei, t. 16 3 (1907), pp. 248-255 and 441-450. 

 t Cf., e. g., Marcolongo, " Teoria Mat. dello Equilibrio dei Corpi 

 Elastici." Milan (1904), p. 233. 



