CHAPTER VIII. 



GRAPHICAL ALGEBRA. 



145. Graphical Mathematics. When the synoptical charts are found which 

 can be derived directly from the observations, the further work for the diagnosis of 

 present or for the prognosis of future states will consist in the performance of 

 mathematical operations with the data given by these charts. The development 

 of proper graphical methods for performing these operations directly upon the charts 

 will be of the same importance for the progress of dynamic meteorology and 

 hydrography as the methods of graphical statics and of graphical dynamics have 

 been for the progress of technical sciences. The first serious problems of these 

 graphical mathematics will present themselves as soon as we shall accomplish 

 kinematic diagnosis by determining the vertical motions. Afterwards we shall meet 

 with such problems continuously. This will therefore be the moment for taking a 

 general view of the character of these problems and of methods to be used for solving 

 them. 



The problems will present themselves in this form : a chart or a set of charts is 

 given, representing the fields of certain scalars or vectors. Another chart or set of 

 charts is to be derived from them, representing the field of other scalars or vectors, 

 which are defined as functions of the first by relations in finite or in infinitesimal 

 form. 



One way for the solution of such problems will always be open. We perform 

 discontinuously, for a certain number of points, the operations defined by the rela- 

 tions. This gives the values of the required scalars or vectors in a certain number of 

 points. By use of these values we draw the charts representing the new scalars or 

 vectors, just as we draw such charts by use of the observations taken at a finite 

 number of points. By following this method we give up the idea of continuous 

 fields during the performance of the mathematical operations, in order to return to 

 the fields as soon as the operations have been performed. We shall call this the 

 discontinuous method. 



But on the other hand it will be possible to find methods by which the idea 

 of the field is never given up. The method will then consist in the continuous 

 tracing of curves guided by the data contained on the given charts, and by the 

 relations containing the implicit definition of the new charts. Every operation leads 

 to a chart representing a field, and it will, as a rule, be necessary to pass through 

 several auxiliary fields in order to arrive at the required fields. We shall call these 

 methods continuous, and the development of them will be our main object. 



146. Drawing-Board. Certain practical arrangements should be mentioned at 

 once. It will be impossible to draw all the different curves on one sheet of paper. 

 They must be distributed on several sheets. But at the same time we must be able 



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