20 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



Now let the curve s be a straight segment of line, and let its length diminish 

 indefinitely. In this limiting case (rr) gives the local rate of variation of the 

 scalar a in the direction determined by the elementary segment of line s. This 

 rate will vary with the direction of s. To examine this variation let us choose the 

 unit of the scalar quantity so small that the thickness of the unit-sheets is small in 

 comparison to the elementary length s. Further, let s have one end-point fixed and 

 let it have a constant length, while it can have any direction. Within the spherical 

 space of radius 5 the equiscalar surfaces separating the unit-sheets can be consid- 

 ered as parallel and equidistant. Then the number of unit-sheets cut by the seg- 

 ment s will evidently be proportional to the cosine of the angle which this segment 

 forms with the normal n to the equiscalar surfaces. The rate of variation of the 

 scalar being in direct proportion to the number of unit-sheets cutting the segment 

 of line s, we get this result: 



The rate of variation of a scalar quantity in any direction s is equal to its rate 

 of variation along the normal to the equiscalar surfaces, multiplied by the cosine 

 of the angle contained between this direction s and the normal n to the equiscalar 

 surfaces. 



17. Gradient and Ascendant. In accordance with this result we can repre- 

 sent the main rate of variation of the scalar field by a vector directed along the nor- 

 mal to the equiscalar surfaces. The rate of variation along any direction is, then, 

 represented by the component of the vector along this direction. The vector may 

 be defined with the positive or with the negative sign, according as the rate of vari- 

 ation be interpreted as the rate of increase, or as the rate of decrease of the scalar 

 quantity. The vector representing the rate of decrease is generally called the 

 gradient and, more specially, potential gradient, pressure gradient, temperature 

 gradient, etc., in accordance with the nature of the scalar quantity. To have a 

 name for the vector representing the rate of increase of the scalar, we shall call it 

 the ascendant. Generally the gradient has the most perspicuous physical sense. 

 But still in some cases the use of the ascendant is to be preferred for practical 

 reasons. 



From what precedes it will be seen that the gradient G and the ascendant A 

 of the scalar a may be defined by the equations 



w *~* 



n being the normal to the equiscalar surfaces, counted positive in the direction 

 of increasing values of a. In the same way the components G s and A s of these 

 vectors along any direction s are given by the rates of decrease or of increase 

 respectively along the direction s 



w -- 



