DIRECT DRAWING OF THE LINES OF FLOW, ETC. 47 



It is important to observe the remarkable completeness of the graphical repre- 

 sentation of a scalar function. When we draw the equiscalar curves for unit inter- 

 vals, these curves will represent not only the scalar itself, but also its ascendant or 

 its gradient (section 17). Any one of these vectors gives complete information 

 regarding the result of any differentiation of the first order performed upon the 

 scalar. The drawing of the equiscalar curves involves therefore a differentiation of 

 the scalar function. The representation gives not only the function itself, but also 

 its differentials. We shall derive great advantage from this later, when we have to 

 perform differential operations in graphical form. 



127. The Drawing of Vector-Lines. The drawing of vector-lines by the use 

 of arrows representing observed directions of a vector and the drawing of equiscalar 

 curves by the use of observed values of a scalar are analogous operations, inasmuch 

 as interpolations have to be performed by eye-measure. But in one case the inter- 

 polations are of scalar nature, in the other of vector-nature, interpolations of 

 direction. 



This difference regarding the nature of the interpolations is intimately related 

 to a difference of principle between the two operations : The drawing of equiscalar 

 curves involves a differentiation in graphical form of a scalar function ; the drawing 

 of the vector-lines involves an integration in graphical form of a differential equation, 

 namely, the differential equation for the vector-curves. We have performed the 

 corresponding analytical integrations in special cases above (sections 120, 122, 123). 

 This graphical integration would not contain any difficulty if arrows of absolutely 

 correct direction completely covered the plane of the drawing. But the curves have 

 to be drawn by the use of the minimum of data given by the observations, and with 

 attention paid to the limited accuracy, or to the direct errors of the observations. 

 Under these circumstances, in order to get the lines drawn as correctly as possible, 

 it will be important to make as complete use as possible of the general properties of 

 the field. We must derive from them qualitative rules which allow us to make the 

 correct use of the data contained in the observations. 



For this we shall have to pay special attention to the singularities of the field, 

 i. e., to the mutual intersections and touchings of the lines of flow; for as soon as 

 the places are determined where intersections or touchings take place, and as soon as 

 the manner is known in which the lines of flow pass through these places, the general 

 feature of the field will to a great extent be given ; for everywhere else in the field 

 the lines will be limited in their course by the condition of not cutting or touching 

 each other. 



128. Simplest Singularities in the Field of the Lines of Flow. We have 

 chosen our examples in the preceding chapter so as to illustrate the simplest 

 singularities which can arise in the three-dimensional solenoidal field ; and forming 

 the horizontal sections through these fields we have seen the character of the corre- 

 sponding singularities in the two-dimensional vector-fields which we shall use to 

 represent the three-dimensional one. In the simple cases treated analytically, the 

 fields had simple properties of symmetry. Drawing correspondingly crooked and 



