CHAPTER IX. 



GRAPHICAL DIFFERENTIATION AND INTEGRATION. 



162. Different Forms of the Problems. We shall meet with problems of 

 graphical differentiation in a variety of forms, each requiring the development of 

 special methods and auxiliaries. The problems will take different forms according 

 as space or time derivations should be performed. The pure space-derivations will 

 depend upon measurements performed upon a chart which represents the given 

 field at a given epoch. The time derivations will involve a comparative investiga- 

 tion of two charts which represent the fields of the same quantity at two different 

 epochs. We shall consider first the space-derivations and afterwards the time- 

 derivations. 



The space-derivations will present themselves in different forms, requiring 

 different methods and auxiliaries according as they depend upon the measurement 

 of lengths or of angles. We shall consider first the angular or directional and then the 

 linear differentiations. To each problem of the differentiation will correspond a 

 problem of integration which in the elementary cases will cause no difficulty as soon 

 as the problem of differentiation is solved. 



163. Directional Differentiation and Integration. Let a system of curves 5 be 

 given; by their tangents they define a system of directions. It is required to find 

 the angle <p which gives the direction of the tangents, i. e., we shall draw the 

 isogonal curves which represent the field of this angle. Evidently this is a problem 

 of differentiation which is inverse to the problem of integration, consisting in the 

 drawing of the vector-lines to a given system of isogons. This drawing of the isogons 

 to a given system of curves can be performed with a certain degree of precision by 

 eye-measure, but a simple auxiliary instrument will be of great use. A transparent 

 circular sheet (fig. 80) can slide in a ring, which has the divisions o to 63 or a 

 certain number of these divisions. On the sheet is drawn a diameter and a set of 

 lines parallel to it. Millimetric distance between them will in most cases be con- 

 venient. The ring is guided so that it has invariable orientation relatively to the 

 system of coordinates. Thus if cartesian coordinates are used, the ring is guided so 

 that it can perform any motion of translation without rotation. In the case of our 

 charts in conical projection the ring is attached to the rule R, which always passes 

 through the point of convergence of the meridians. (Compare fig. 62.) The sheet 

 has a small perforation at the center, which allows us to mark the points where the 

 desired values of the angle are found. 



The sheet is guided in such a way that its center (the hole) follows one of the 

 given curves. During the displacements it is turned so that the diameter remains 

 tangent to the curve. The adjustment to tangency will be very much assisted by 

 the lines which are parallel to the diameter. Whenever the diameter points to 



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