864 THE SPIEAL SHELLS [ch. 



still; but from the symmetry of the case and the continuity of the 

 whole phenomenon, we are entitled to believe that the conditions 

 are just the same, or .very nearly the same, time after time, from one 

 chamber to another : as the one chamber is conformed so will the next 

 tend to be, and as the one is situated relatively to the system so will 

 its successor tend to be situated in turn. The physical law of minimum 

 potential (including also the law of minimal area) is all that we need 

 in order to explain, in general terms, the continued similarity of one 

 chamber to another; and the physiological law of growth, by which 

 a continued proportionality of size tends to run through the series 

 of successive chambers, impresses the form of a logarithmic spiral 

 upon this series of similar increments. 



In each particular case the nature of the logarithmic spiral, as 

 defined by its constant angle, will be chiefly determined by the rate 

 of growth ; that is to say by the particular ratio in which each new 

 chamber exceeds its predecessor in magnitude. But shells having 

 the same constant angle (a) may still differ from one another in many 

 ways — in the general form and relative position of the chambers, 

 in their extent of overlap, and hence in the actual contour and 

 appearance of the shell; 'and these variations must correspond to par- 

 ticular distributions of energy within the system, which is governed 

 as a whole by the law of minimum potential. 



Our problem, then, becomes reduced to that of investigating the 

 possible configurations which may be derived from the successive 

 symmetrical apposition of similar bodies whose magnitudes are in 

 continued proportion; and it is obvious, mathematically speaking, 

 that the various possible arrangements all come under the head of 

 the logarithmic spiral, together with the limiting cases which it 

 includes. Since the difference between one such form and another 

 depends upon the numerical value of certain coefficients of mag- 

 nitude, it is plain that any One must tend to pass into any other 

 by small and continuous gradations; in other words, that a classi- 

 fication of these forms must (like any classification whatsoever of 

 logarithmic spirals or of any other mathematical curves) be theoretic 

 or "artificial." But we may easily make such an artificial classi- 

 fication, and shall probably find it to agree, more or less, with the 

 usual methods of classification recognised by biological students of 

 the Foraminifera. 



