178 Dr. 0. Chree on Applications of Physics 



This holds true of V and w all over the" surface, and so 

 applies likewise to their differential coefficients with respect to 

 #o and ?/ — so far at least as concerns points, outside the loaded 

 area. 



The direction-cosines of the normal to the surface after the 

 application of the load are, to a first approximation, 



div dio 

 dx ' dy ' 



so that the slope at the point # , y Q is given by 



*-{©)■*©»' «> 



y 



of gravity from 



Again, to a first approximation the presence of the loading- 

 material has altered the direction-cosines of the line of action 



1 dY 1 dY 

 0, 0, 1 to — j — , - -j — s 1. 



Thus gravity has become inclined to the vertical at the angle 



*.-H(SHf:)T- •■■<»> 



Employing (6) in (7) and (8), we obtain the elegant 

 relation 



tV*2=(l-^)//(2™y). .... (9) 



The spirit-level measures ^i + ^2? which always exceeds the 

 true change of level tyi. 

 Since 



dwo j dwo _ dVo / ^Vq ^ 



dw j dy " dx \ dy Q ' 



the final directions of gravity and of the normal to the surface 

 lie in the same vertical plane ($. e. plane through z). This 

 result may facilitate experimental investigations, as a rough 

 idea of the direction of the resultant attraction of the loading- 

 material will generally be obtainable by eye. The possible 

 influence of want of symmetry in the contour of the ground, 

 or variability of the surface-strata, must of course be borne 

 in mind. 



The relation (9), so far as I know, is new. Its discovery 

 was due to a faint impression that a formula I had obtained 

 for the effect of a loaded rectangle resembled something I 

 had s@en° before, the something proving on investigation to be 

 result (7) in Thomson & Tait's Nat. Phil., art. 818. 



