188 Dr. C. Chree on Applications of Physics 



isolated load pdco at a point P on the surface, and consider 

 the vertical displacement at a point Q at depth z. Join 

 QP = r, and draw QM = ^' perpendicular on the vertical PM. 

 Then denoting the angle QPM by a, we have at Q 



tv = (pdco/4:7rnr){2{l-v)+cos 2 a}, (32)* 



= Ow7fi>/4wW) {2(1-17X1 + z 2 /.v ,2 )'i+ (z/x') 2 {l + 2 /a?' 2 )-*}. 



Thus when z/af is small, we find, neglecting powers of zjx' 

 above the second, 



w=(pda>/27rnx / ){l- V + ± v {z/x f ) 2 }; . . . (33) 

 whence 



-^ = (pdco/2Trnx>*){l-r, + i V (z/*;r}- ■ ■ $i) 



Here the slope — dwjdx 1 increases at first with the depth as 

 in the case of (30) and (31). 



§ 20. When the depth is of the same order of magnitude as 

 the horizontal distance of the nearest point of the loaded area, 

 individual cases of (14) or (29) require separate consideration. 



When the depth becomes large compared to the horizontal 

 distance of the remotest point of the loaded area, we easily 

 find from (14) as a first approximation 



(dwV^ _ (5-2 V )p(y 2 -y 1 )(x 2 2 -w 1 2 ) 



\dx) z=z ~ 87m "z^~ ~ ' W 



showing that the slope now diminishes as the inverse cube of 

 the depth. 



If P denote the total load, a, y the coordinates of the C.Gr. 

 of the loaded area, we have at once from (35) 



(dw\ x=z v =0 

 &L, =( 5 -^)p*/( 4 ™ 3 )> • • • ( 36 ) 



and by symmetry 



^L =(5-2*?)Py/(W 3 ). . . . (37) 



The line of greatest slope is thus in the vertical plane which 

 contains the C.Gr. of the loaded area, and if R be the hori- 

 zontal distance of the CGr., the slope is given by 



(dw/dR) z=z =(5-2 v )?'R/(4>7rnz d ). . . . (38) 



The conditions assumed in (35) are practically tantamount to 

 those of the elementary loaded area, and (38) can in fact be 

 deduced from (32) by supposing a small. 



* Todhunter & Pearson's ' History/ vol. ii. eqn. (xxiv.) of art. 1497, 



