28 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 5 I 



Imagine the three coordinate axes movable with the earth to be 

 so drawn through this point that the xy plane is horizontal, x being 

 positive eastward along the tangent to the small circle of latitude 

 and y positive northward along the tangent to the meridian but z 

 positive downward toward the center of the earth considered as a 

 sphere. Let 01 be the direction of the positive half of the momen- 

 tary axis of rotation or in our case a line drawn through parallel 

 to that half of the earth's axis that extends from its center to the 

 south pole; then in general 



to l = to 2 cos (Ix) to 2 = — to cos (ly) to 3 = to cos (Iz) 



and in our special case 



to x = to to 2 = — to cos X to 3 = to sin X 



where X is the latitude of the place of observation (O) and to is the 

 angular velocity corresponding to the diurnal rotation of the earth. 

 Since in this case of steady rotation 



• dvi = d&2 = dc °3 = 



dt dt dt 



and considering the conditions expressed in equation (3), therefore 

 in the present problem the general equations (2) become 



<Px x d 2 x I dz dy\ . 



— - = — — 2 tot cos X - - + sin / — to 2 . x 



dt 2 dt 2 \ dt dt I 



u?'V d?'V doc 



— = — ~ — 2 to sin X — — to 2 cos X ( — cos X.y + sin X.z) — afy I to \ 

 dt 2 dt 2 dt 



c*z d^z doc 



— - = — 2 to cos X - — to 2 sin X ( — cos X.y + sin X . z) — to 2 z 



dfi dt 2 dt 



The terms in to 2 may be omitted because they are very small; but 



dz 



for the same reason the term 2 to cos X . must be omitted in all those 



at 



cases in which the gradient of the surface (the rails or the river), 



and hence also — , is small or zero. 

 dt 



Let a be the angle between the direction of the motion of the point 



x, y, z and the direction of the positive axis of y and let v be the 



