908 DR JOHN DOUGALL ON AN ANALYTICAI. THEORY OF THE 



right of 0, when z>z'\ when 3<2', vve may either change to the series of residues at 

 and at the poles of D,„ to the left of 0, or, preferably, change the variable from ^ to 



— p m (2), wliicli has the same enect m this case as changing e « into —e~or. 

 Thus we find yor a unit elerneid of' longitudinal traction at (a, w\ z') 



/C> C\ e,„ „ 1 ( A3,„,B, „,,»-, /ip -«i^' COS , ,, 



\ "A/.^. J TT/ji^fSdDJdf^] G,,,„ J "'a «iii ^ ^ 



e,„ ui • . r 1 ■ 1 ( A.. B.. I ^ /3p -5^^:^ cos , 



- — !^ . coethcieut of ^ in -—— ' •'■ "•' "■^- '" ' I J ^e a . ■tnico - oj ), 



(3) 



for z>z'; for 2;<2', interchange z and 2' and change the signs of (^, 0, and ^. 



For a P or an ^ element of traction, (/), 0, and v|/ are even functions of 2 — 2' ; for a 

 Z element they are odd functions of 2 — 2'. 



The solutions for the wi-components of an element of traction at {a, co', z') may be 

 considered to belong to a linear distribution of traction over the circle 2 = 2', p = a, the 



intensity of which is -^ cos m{co — w') per unit length. 



7ra 



9. Simjdifications in the case of traction s ipmnetrical about the axis. 



The solutions given in Arts. 5 to 8 do not fail when 'm=0, but on account of the 

 vanishing of «», ^-^d 62 0, the determinant Dq reduces to C2_oC.2^o- 



The effect of this is that the determination of v/^ becomes independent of that of 

 ({) and 6 ; also that for normal and longitudinal traction only <p and occur, for 

 transverse traction only \//-. 



The simplified results may easily be deduced from those of the general case, or 

 they may be investigated independently. We merely write them down here. 



In each case what is given is the symmetrical comjwnent in the solution for a unit 

 element of traction. 



Normal traction. 



|3(Z-2') ^ 



<*• ^> - - ^^iijsf<'''' "*'■ " "'">''"^" 



«2-/) 



1 11 «o -P^-"^' 



---L . coefficient of L i,, ^(2^8-)', v/3J' - 2/32J),l„/^e « , 



for 2>2' ; for 2<2', interchange 2 and 2' in </> and 0. 



The summation extends over the zeroes of C2, u with positive real part. 

 Transverse traction. 



JLV 1 J lip -g^> 



1 \ 1 Bp -^-^^^"> I ' ' ' " 



- - — . coefficient of - in — -— ^„^ „„ Jn— e " ) 



for z'^z' ; for 2<2', interchange 2 and 2' in ^. 



The summation extends over the zeroes oi /SJ^/S — ^^J^'^ with positive real part. 



(1) 



(2) 



