IOO 



DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



one of the integer divisions on the ring we make a mark on the curve through the 

 hole. In this manner the disk is guided along the given curves, and marks are 

 made where the required isogonal curves should intersect them. Afterwards these 

 isogonal curves can be drawn continuously. If they are made to pass precisely 

 through the points marked they will always show oscillations in their course, due to 

 the unavoidable errors accompanying the drawing of the given curves and the use of 

 the differentiating instrument. But these irregularities are easily smoothed out 

 on the final drawing of the curves. 



It will be important to remember that the curves which we obtain by this 

 instrument can be numbered so as to be the isogons of the curves 5 themselves, 

 or so as to be the isogons of the curves which are normal to the curves s. We pass 



Fig. 80. Divided sheet for directional differentiation. 



from the isogons of the curves 5 to those of their positive normal curves by an addi- 

 tion of 1 6, to those of their negative normal curves by an addition of 48 to the 

 numbers which the isogons have when they represent the curves s. 



We have treated already (Chapter VII) the problem of integration which is 

 inverse to the directional differentiation. Evidently the sheet of fig. 80, by which 

 we perform the differentiations, may also be used to assist the integrations; and 

 the integration-machine of fig. 62 or 63 can be considered as intrinsically the same 

 instrument as that of fig. 80, only provided with special devices for facilitating the 

 practical work connected with the integration. 



164. Linear Differentiation and Integration. Let the scalar a have a definite 

 value at every point of a line s; i. e., let a be a function of the length of arc 5 

 (a) a = a(s) 



