Connected with the Earth's Field of Force. 153 

 for the origin this gives 



L^ 9xJo \-.9x9z 9zj: 



or since 



we get 



[£] 



= -9o 









\-9x9zJq 



But 



LP a; Jo L9<l>9xAo R L5<^Jo' 



where ^ is the latitude. As in Appendix A 

 ||=^^^sin2^, 



where ge denotes gravity at the equator and jS is a constant 

 of the gravity formula = 0.00529 nearly. With sufficient 

 accuracy we may substitute g^ for ^ so that we get finally 



. 2 h =^pshi2cl> (4) 



The component of pressure along the rr-axis, px, is 

 evidently given by 



p^ — — I I p cos a dS, (5) 



where p is the pressure on the element of dB of the sub- 

 merged surface, and a is the angle between the external 

 normal to the surface and the :r-axis; the integration 

 extends over, the entire submerged surface. If we take 

 points in pairs like P and Q in the figure, the two values 

 of cos a are numerically equal but opposite in sign, and 

 by writing (V — Vo) p for the pressure, noting that cos a 

 d8 = dy dz and using V with subscripts P and Q to indi- 

 cate the values of V at these points, Ave find 



P. = -ff (Vp - YJ 9 dy dz. (6) 



The integration in (6) extends over a section of the 

 sphere by the ;z/s-plane, the bounding curves being the 

 projection of the ''water-line" on the ^/^-plane and that 



