u,„ = - - — . \ r^e a, dp, (third and fourth paths), ..... (2) 



948 DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



The w-component of the value of any displacement u will be given in the form, say, 



1 /"U _££::£) 

 u„,= --J^\^e a fl!/^, (third and fourth paths) (1) 



Here U„,/D,„ is a uniform function of /3, even or odd according to the directions of 

 the force, and of the displacement m. Thus a differentiation as to z or z' in the 

 application of 15 (17) and 2 (7) introduces (i in the first power, but a differentiation 

 as to jO, p, o) or w' leaves an even or odd function still even or odd. But <p in 

 16 (10) and the corresponding forms for 6 and \|/ involve even functions of |S as factors 



of e~ « , so that in (1) U,„/D„, is even in /3 for the displacements u^ and u^ in the 

 case of a P or an O force, and for the displacement u^ in the case of a Z force ; in the 

 other cases 11^/0^ is odd in /3. 



When U,„/D„, is even, (1) can also be written 



4772.' 



but when U^/D,„ is odd, 



1 f\j ^"-- ^ 



u„, — + — ^e a clB, (third and fourth paths). . . . . (3) 



4:TnJ L),,„ ' 



In all cases U,,^ is the product of l/fxa by a function of pja and p'/a (and of w, w'), 

 that is to say, it is the product of 1/m" by a function of zero dimensions in the unit 

 of length and of order zero in a, when a is regarded as infinitesimal. 



For definiteness we shall now take a P force and the displacement u^. 



The transformation of (1) and (2) by Cauchy!s Theorem gives, as at 6 (9), 



for z>z', 



1 1 U _&lz^ "V u /3(z-z') 



Wm = o coefficient of - in -^'e a + Zt T.^ e « , . • . . (4) 



2 /3 1)„, ,3 dDJd.p ' 



for z < z, 



1 1 U ^(^-^') -sri TJ ^^'^'> 

 u,„ = - coefficient of - in — ^e a + /_, '!^ — a a . . . . . (5) 



In the integrand of (l) {cf. 16 (10)), the form of U,„ varies according as jO> or 

 <jo', but, now that we are dealing with forces and displacements, the terms which 

 alter have no residue at /3 = or at any other zero of D,„, so that no change is 

 produced in (4) or (5) by the interchange of p and p {cf footnote. Art. 17). 



Pass now to the problem of a body distribution of P force of intensity P(jo', w , z') at 

 (p', M , z). There will be a solution in which the m-component of the value of u^ is 



^P,m= P'(^'\ <^^' \ ^{P, w', z')u„,dz', (6) 



where u^ is the function defined in (4) and (5). 



(We suppose throughout that the applied force vanishes unless z lies between 

 Zi and 22, where 22>^i-) 



Consider in the first place the integral as to z' in (6), viz., 



I = j 1*(f',1w', z')u„,dz (7) 



