180 Dr. C. Chree on Applications of Physics 



The corresponding approximation to the slope, viz., 



-aE=(i-9)-r/(*«B*), . . . (12) 



shows that at considerable distances from a small loaded area 

 the slope varies approximately as the inverse square of the 

 distance. In (11) and (12) the distribution of load is not 

 assumed uniform. 



The fact that (11) holds only when the distance of the 

 loaded area is so large that its effect is relatively small 

 diminishes its value in practice. 



§ 9. The determination of w from (3) entails the evaluation 

 of two integrals, neither very manageable. For points on 

 the surface there is, however, only the single integral (4). 

 This has been converted by Boussinesq into two alternative 

 forms — one for points outside, the other for points inside the 

 loaded area — which are convenient when the load, though 

 not necessarily uniform, is distributed symmetrically round a 

 point (see Todhunter & Pearson's ' History,' arts. 1501 

 and 1502). In this way the depression can be easily deter- 

 mined at the centre and edge of a circular depressed area for 

 a variety of laws of loading, and the depression at other 

 points can be expressed in terms of infinite series or elliptic 

 integrals (see Todhunter & Pearson, I. c, especially art. 1501). 



The slope in these cases, at any distance from the centre of 

 the loaded area, can be obtained in the form of infinite series 

 or elliptic functions ; but results of this kind are more apt 

 to repel than to enlighten the unmathematical reader. 



Fortunately, when the load is uniform, and the loaded 

 area rectangular, it proves possible to express the components 

 of slope dw/dx, dw/dy at any point of the surface in terms of 

 ordinary Napierian logarithms. I shall accordingly devote 

 attention almost exclusively to this case. 



§ 10. Returning to (3), let x f , y' be coordinates of the 

 element dco, so that 



day = dx'dy', 

 f*=(«— a/)* + (y-y)«+jA 



The loading being supposed uniform, we have 



But 



dx \l) = " dx' W' and dx \?) = " drf\?) ; 



