GRAVITY AND GRAVITY POTENTIAL. 21 



The equiscalar surfaces or the unit-sheets ? representing the field of the 

 scalar quantity a give at the same time a complete representation of the field of the 

 vector G or A. From what is stated above we can immediately draw these 

 conclusions: 



(i) The direction of the vectors is that of the normal to the equiscalar surfaces. 



(2) If a sufficiently small unit be used, the magnitude of the vector will be repre- 

 sented numerically by the number of unit-sheets per unit-length of the normal; or, 

 what comes to the same thing, by the reciprocal thickness of the unit-sheet. 



(3) The component of any of the vectors in any direction s is numerically equal 

 to the number of unit-sheets per unit-length in this direction; or, in other words, 

 it is equal to the reciprocal length of that segment of the line s which is contained 

 in a unit-sheet. 



If, finally, we add that the gradient points in the direction of decreasing and the 

 ascendant in the direction of increasing values of the scalar, we see that the 

 equiscalar surfaces and the unit-sheets give a full representation of the field of the 

 gradient or of the ascendant. If greater units be used, so that the unit-sheets 

 have greater thickness than supposed above, perfectly corresponding theorems may 

 be formed for the average values of the vectors or their components referred to 

 definite lengths of the segment s. 



As we can pass by a process of differentiation from the field of a scalar to the 

 field of its gradient or its ascendant, we can, vice versa, return by a process of 

 integration from one of the latter fields to the first. To show this, say tor the 

 gradient, we can multiply equation (c) by the line element ds and integrate along 

 the curve s from a point o to a point 1. This gives 



() G.ds=-f o P s d S = -da= ait -a l 



and ! being the values of a at the points o and 1, respectively. The first member 

 of this equation is the line-integral of the component of the vector G tangential to 

 the curve s. As we shall usually have to take line-integrals only of the tangential 

 vector-components, we may denote an integral of this nature simply as the litie- 

 integral of the vector. This line-integral of the gradient gives us the means of 

 reconstructing the field of the scalar. For, knowing the field of the gradient and 

 the value of the scalar quantity in one point of space, we can find the value of the 

 scalar in any point by integrating the gradient along any curve leading from the 

 first point to the second. 



It will be useful, finally, to express in terms of the gradient the ratio (a), section 

 16, from which we derived originally the definition of this vector. Taking in the 

 integral (e) the mean value G sm o{ the tangential component of the gradient outside 

 the integral sign, the integration can be performed, and gives the length s of the 

 curve. Dividing by this s, we get 



CO 6, 



S, 771 



Thus, the mean value of the component of the gradient tangential to any curve s is 



