950 



DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



derivatives. These functions are defined in terms of the given applied force. The 

 process will be illustrated by application to the first group of terms in Art. 25, and 

 the results for the complete set of terms will then be written down. 



The first line in the table of Art. 25 gives, after integration through the volume 

 and over the surface : — 



if 2 < 2j , 



rEa* .u^= - — (z - 2')32X(.x'', y', z')dz' - (o-.r^ - a-y^ - ra^) / (2 - 2')2X(x', y', z)dz ; 



if 2] <2<Z2, 



TrEa* 



if 2>22, 



.u,= f^ 1(2 - zy-EXix', y', z)<lz + ((7x2 _ ^y2 _ ^a2)jr^ (, _ 2')2X(a;', y\ z)dz | ' 

 - j["'l(^ - 2T2X(a;', y\ z')dz - (ax' - cry^ - ra2)^''(z - z)^X{x', y' , z')dz' ; 



(3) 



ttEo* . u, = t\-{z - 272X(a;', ij, z')dz' + (o-x^ _ ay"- - to^) hz - 2')2X(a;', y', z')dz . 



Let f{z) be an integrable function of z which vanishes unless 2i<2<22, and let 

 a function x{^) be defined for all values of z as follows, n being a positive integer : — 



if 2 < 2i , x(^) = - [\'' - 2')7'(2>?^' ; 



if 2i < 2 < ., , x(2) = [ (^ - ^rf{^')dz' - j\z - 2')y(2>2' ; 

 if 2 > 22, x(^)= \\z-z'Yt\z')dz'; 



or, more succinctly, for all values of z, 



X{z) = l\z - zr/iz')dz - j\z - zrf{z)dz. . . 



(4) 



(5) 



Then 



'L 



dz 



X(2) = «£ (2 - 2')"->/-(2>iz' - « j % - Zy~'f{z)dz', 



^.xi:^ = KnUz). 



(6) 



For the function ~. — yXi^)^ where x{^) i^ defined as in (4), we shall use the 



2(n !) 



notation 



Dr"-y(2). 



Thus, if r is a positive integer, 



£,Dr-y(2)=Dr"-^+7(2), 

 |^D-"-y(2)=/(2), 



(7) 



(8) 



and D ''f{z) is continuous for all values of z. 



