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DARCY WENTWORTH THOMPSON ON 



(2) The second type is that where extension is not equal or uniform at all 

 distances from the origin : but grows greater or less, as, for instance, when we stretch 

 a tapering elastic band. In such cases, as I have represented it in fig. 4, the 

 ordinate increases logarithmically, and for y we substitute e y . It is obvious that this 

 logarithmic extension may involve both abscissae and ordinates, x becoming e x , while 

 y becomes e y . The circle in our original figure is now deformed into some such shape 

 as that of fig. 5. This method of deformation is a common one, and will often be of 

 use to us in our comparison of organic forms. 



(3) Our third type is the " simple shear," where the rectangular co-ordinates 



Fjo. 5. 



Fift. 6. 



Fig. 7. 



become " oblique," their axes being inclined to one another at a certain angle ». 

 Our original rectangle now becomes such a figure as that of fig. 6. The system may 

 now be described in terms of the oblique axes X, Y ; or may be directly referred to new 

 rectangular co-ordinates £, 1 by the simple transposition x=^—*i cot «>, y—>i cosec "\ 



(4) Yet another important class of deformations may be represented by the use of 

 radial co-ordinates, in which one set of lines are represented as radiating from a point 

 or "focus," while the other set are transformed into circular arcs cutting the radii 

 orthogonally. These radial co-ordinates are especially applicable to cases where 

 there exists (either within or without the figure) some part which is supposed to sutler 

 no deformation. A simple illustration is afforded by the diagrams which illustrate 

 the flexure of a beam (fig. 7). In biology these co-ordinates will be especially 

 applicable in cases where the growing structure includes a " node," or point where 



