1904-5.] Prof. Chrystal on Mathematical Theory of Seiches. 645 
The truth, however, is that the lakes, such as Ness and 
Geneva, where Du Boys’ rule gives the best results, have normal 
(<r- v-) curves that are neither purely concave nor purely convex, 
hence the errors of the formula compensate one another; and 
the result is by an accident nearer the truth than the vague 
theoretical basis of the formula would lead us to expect. 
Inasmuch as very pronounced convexity is an uncommon 
feature of lakes in nature, we should expect that Du Boys’ rule 
would never give the period in concavo-convex lakes much under 
the true value. In purely concave, or in concavo-convex lakes, 
where the concavity in the normal curve greatly predominates, 
Du Boys’ rule may give a result considerably over the truth. 
Thus, for example, for a purely parabolic symmetric lake 
d T i = {%/J(gh)} f _dxa/J(a 2 - x>) = irl/ J(gh), 
whereas Tj = irlj J(2gh) by the theory I have developed. Hence 
dTj/Tj = = 1*414. Nevertheless, Du Boys’ rule is valuable 
as an empirical formula, easy of application, and giving in many 
cases a good first approximation, which is better the larger the 
number of nodes. It must also retain a historical interest as one 
of the first successful attempts to submit the very complicated 
phenomena of seiches to calculation. 
Anomalous Seiche in a Concave Quartic Lake. 
§ 16. We now return to consider the solution of the differential 
equation (8) for which 0 = 0, and therefore c = a 2 . The roots of 
the quadratic (10) are then equal; and the process of § (5) gives 
only one independent fundamental integral, viz. : — y = (a 1 - x 2 )“. 
The solution can, however, he completed by a well-known use of 
this particular integral. We thus get 
P = (a 2 -x 2 )‘<j A + Blog^±^ l. . . (47) 
If a = log {{a + p)/(a - p) , ft = log {(a + q)/{a - ?)}, the 
boundary conditions £= 0 , when x = p and x = q , would require 
A + Ba = 0, A + B/3 = 0 . . . (48) 
From (48) it would follow either that a = j3, or that A and B 
