; BILLS AND DALES. 



with sufficient accuracy by the number of feet above the mean level of the 

 M*; but when the survey is so extensive that the variation of the force of 

 gravity must be taken into account, we must adopt a new definition of the 

 height of a place in order to be mathematically accurate. If we could deter- 

 mine the exact form of the surface of equilibrium of the sea, so as to know 

 its position in the interior of a continent, we might draw a normal to this 

 surface from the top of a mountain, and call this the height of the mountain. 

 Tins would be perfectly definite in the case when the surface of equilibrium 

 is everywhere convex ; but the lines of equal height would not be level 

 surfaces. 



Level surfaces are surfaces of equilibrium, and they are not equidistant. 

 The only thing which is constant is the amount of work required to rise from 

 one to another. Hence the only consistent definition of a level surface is 

 obtained by assuming a standard station, say, at the mean level of the sea 

 at a particular place, and defining every other level surface by the work 

 required to raise unit of mass from the standard station to that level surface. 

 This work must, of course, be expressed in absolute measure, not in local foot- 

 pounds. 



At every step, therefore, in ascertaining the difference of level of two 

 places, the surveyor should ascertain the force of gravity, and multiply the 

 linear difference of level observed by the numerical value of the force of gravity. 



The height of a place, according to this system, will be defined by a 

 number which represents, not a lineal quantity, but the half square of the 

 velocity which an unresisted body would acquire in sliding along any path from 

 that place to the standard station. This is the only definition of the height 

 of a place consistent with the condition that places of equal height should be 

 on the same level. If by any means we can ascertain the mean value of 

 gravity along the line of force drawn from the place to the standard level 

 surface, then, if we divide the number already found by this mean value, we 

 shall obtain the length of this line of force, which may be called the linear 

 height of the place. 



On the Forms of Contour-lines. 



Let us begin with a level surface entirely within the solid part of the 

 earth, and let us suppose it to ascend till it reaches the bottom of the deepest 



