Solid Cylinders of Elliptic Section. 155 



in the length of the cylinder is zero*. It seems, however, 

 unnecessary to reproduce the algebraical work of determining 

 the arbitrary constants occurring in my general solution of 

 the elastic solid equations. The work is of no interest in 

 itself, and the accuracy of the solution may be easily tested 

 by reference to the internal equations and surface conditions 

 it has to satisfy. It satisfies exactly the internal equations 

 (3), the three conditions (7), (8), and (9) over the curved 

 surface, and the conditions (4) and (5) over the flat ends. 

 There remains only the last surface condition over the ends, 

 z = +lj viz, 



If this were satisfied the solution would be absolutely exact, 

 and applicable however great or small l/a might be. This 

 condition is satisfied when v~0, but otherwise we have to 

 avail ourselves of the principle of statically equivalent load 

 systems, replacing the above condition by 



U 



a Va ,<J -x" 2 



zz dxdy = (75) 



a ya 2 —z 2 



This equation the following solution will be found to satisfy 

 for every cross section, and not merely for the ends. The 

 notation is the same as in Part I. 



E A (1 - ri) (3a 4 + 2aW + 3b*)/<D*p = 



(l-27;)[(a 2 + ^){a 4 + a^HM-M 3 ^- 2 ^ 2 + 3 ^)-i^V-^) 2 } 

 -(l+rj)x*{a± + a%* + 2V-r 1 (at-V))-(l + r l )if (76) 



Ea(l - V ) (3a 4 + 2a 2 £ 2 + 3b 4 )/a)*p = 



x{a%a 4 + a% 2 + b 4 ) - V (a 6 + a 4 b 2 + 2a% 4 + b 6 ) -\rf(ti 2 -b 2 )(a 4 + ?>b 4 ) 



-i(l+V>^a 4 + a 2 b 2 + b 4 -7 ] (2a 4 + a% 2 + 3b 4 )+r ] \a 4 -b 4 )} 

 -(H-77)^ 2 {a 4 -^ 2 (a 2 + 5 2 )-7; 2 (a 4 -^ 4 )}, (77) 



* In this case the solution is the same as for the thin disk, and also as 

 equations (131)-(133) on pp. 31-32 Quarterly Journal, vol. xxiii., when 

 m is put =n, and an obvious printer's error interchanging x and x 3 in 

 (131) is corrected. 



