730 THE THEORY OF TRANSFORMATIONS [ch. 



into an ellipse. In elementary mathematical language, for the 

 original x and y we have substituted x-^ and cy-^, and the equation 

 to our original circle, a;^ + ^^ = a^, becomes that of the ellipse, 

 Xy + c^yx == a^. 



If I draw the cannon-bone of an ox (Fig. 354, A), for instance, 

 within a system of rectangular co-ordinates, and then transfer 

 the same drawing, point for point, to a system in which for the 

 X of the original diagram we substitute x' = 2a;/3, we obtain a 

 drawing (B) which is a very close approximation to the cannon- 

 bone of the sheep. In other words, the main (and perhaps 

 the only) difference between the two bones is simply that that of 

 the sheep is elongated, along the vertical axis, as compared with 

 that of the ox in the relation of 3/2. And similarly, the long 

 slender cannon-bone of the giraffe (C) is referable to the same 

 identical type, subject to a reduction of breadth, or increase 

 of length, corresponding to x" = x/3. 



(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. 355, the ordinate 

 increases logarithmically, and for y we substitute e^. It is obvious 

 that this logarithmic extension may involve both abscissae and 

 ordinates, x becoming e^, while y becomes e^. The circle in our 

 original figure is now deformed into some such shape as that of 

 Fig, 356. 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 become "oblique," their axes being inclined to one 

 another at a certain angle co. Our original rectangle now becomes 

 such a figure as that of Fig. 357. The system may now be 

 described in terms of the oblique axes X, Y ; or may be directly 

 referred to new rectangular co-ordinates $, -q by the simple 

 transposition x = $ ~ t] cot co, y = rj cosec oj. 



(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 



