XVII] THE COMPARISON OF RELATED FORMS 1039 



in the original "Cartesian" net will now, after extension in the 

 ^/-direction, be found elongated into an ellipse. In elementary 

 mathematical language, fo;- the original x and y we have substituted 

 iCj and c?/i , and the equation to our original circle, x^ ^ ip' = a^, 

 becomes that of the ellipse, x^ + c^yx ^ ^^• 



If I draw the cannon-bone of an ox (Fig. 493, A), for instance, 

 within a system of rectangular coordinates, and then transfer the 

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

 the original diagram we substitute x' = 2x/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 



A B C 



Fig. 493. 



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 pro- 

 portion of 3/2. And similarly, the long slender cannon-bone of the 

 giraife (C) is referable to the same identical t3rpe, 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 sl tapering elastic band. In such 

 cases, as I have represented it in Fig. 494, 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. 495. This 



