where _A_, _C_, and _a_ are certain specific constants, varying with the part of the plant which we happen to be considering. On the main stem, the rate of change in the number of leaves per whorl is very slow; when we come to the small twigs, or “tertiary branches,” it has become rapid, as we see from the following abbreviated table: _Number of leaves per whorl on the tertiary branches of Ceratophyllum._ Position of whorl 1 2 3 4 5 6 Mean number of leaves 6·55 8·07 9·00 9·20 9·75 10·00 Increment — 1·52 ·93 ·20 (·55) (·25) We have seen that a slow but definite change of form is a common accompaniment of increasing age, and is brought about as the simple and natural result of an altered ratio between the rates of growth in different dimensions: or rather by the progressive change necessarily brought about by the difference in their accelerations. There are many cases however in which the change is all but imperceptible to ordinary measurement, and many others in which some one dimension is easily measured, but others are hard to measure with corresponding accuracy. {98} For instance, in any ordinary fish, such as a plaice or a haddock, the length is not difficult to measure, but measurements of breadth or depth are very much more uncertain. In cases such as these, while it remains difficult to define the precise nature of the change of form, it is easy to shew that such a change is taking place if we make use of that ratio of length to weight which we have spoken of in the preceding chapter. Assuming, as we may fairly do, that weight is directly proportional to bulk or volume, we may express this relation in the form _W_/_L_^3 = _k_, where _k_ is a constant, to be determined for each particular case. (_W_ and _L_ are expressed in grammes and centimetres, and it is usual to multiply the result by some figure, such as 1000, so as to give the constant _k_ a value near to unity.) _Plaice caught in a certain area, March, 1907. Variation of k (the weight-length coefficient) with size. (Data taken from the Department of Agriculture and Fisheries’ Plaice-Report, vol._ I, _p._ 107, 1908.)