58 J^rorrrdiiif/s of thp llotjal Irish Acadetny. 



on the exponential (j[mz)^ and therefore decrease very rapidly as we go 

 inward ' from the surface. This case, then, verifies St. Tenant's 

 general theory of equipoUence. 



We may now note specially three cases of distribution of shearing 

 force : — 



I. The vanish, <^(r) being c^r. 



"We have now u = c^yz, v = - CqZx^ lo = 0. 

 This constitutes Coulomb's and St Tenant's solution for circular 

 cylinder under torsion. 



II. The torque over terminal is confined to the neighbourhood 

 of the centre. 



Here 



ni 



-TFT? — '\ r^(f>{r)dr, 



R being radius of activity of terminal torque. 



III. The action of terminal torque is confined to rim. Here the 

 are proportional to 



1 



Ji {ma) ' 



while the 3 , . , 3 



Cq= ^ '^^ v) ^ ^^^^^ torque. 



L, — Consider, now, the case where the equation determining 

 the m is 



t/i {ma) = 0. 



We have now for the curved surface u = v = w = 0^ i.e. the 

 surface is held. The surface tractions which must operate for 

 this purpose are 



X = y^AJ {mz) - {ma), F = - x^^AJ {mz) - J, {ma), Z=0, 

 a a 



i. e. a tangential force ^mA„,8 {mz) . {ma). 



To determine the A„j we have now to express the given law 

 of shearing force over terminal in a series 



cf>{r) = CQr+'^c^J,{mr). 

 We have now as before 



r " r " a^ 



rJ^{mr)Ji{m'r) = 0, rJi^ {mr) dr = — Jq" {ma). 



Jo Jo ^ 



