XII] OF THE FORAMINIFERA (jOl 



In the molecular growth of a crystal, although we must of 

 necessity assume that each molecule settles down in a position of 

 minimum potential energy, we find it very hard indeed to explain 

 precisely, even in simple cases and after all the labours of modern 

 crystallographers, why this or that position is actually a place of 

 minimum potential. In the case of our little Foraminifer (just 

 as in the case of the crystal), let us then be content to assert that 

 each drop or bead of protoplasm takes up a position of minimum 

 potential energy, in relation to all the circumstances of the case; 

 and let us not attempt, in the present state of our knowledge, to 

 define that position of minimum potential by reference to angle 

 of contact or any other particular condition of equiUbrium. In 

 most cases the whole exposed surface, on some portion of which 

 the drop must come to rest, is an extremely complicated one, and 

 the forces involved constitute a system which, in its entirety, is 

 more complicated 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 pro- 

 portionality of size tends to run through the series of successive 

 chambers, impresses upon this series of similar increments the 

 form of a logarithmic spiral. 



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 particular distributions of energy within the system, 

 which is governed as a whole by the law of minimum potential. 



