638 . MESSRS A. T. DOODSON, R. M. CAREY, AND R. BALDWIN, THEORETICAL 
was only able to place them approximately. He places the node of the uninodal 
oscillations somewhere to the west of the line Rolle-Tbonon ; but he also records 
that no uninodal oscillation was observed at Sechex or at Fleur D’Eau. Our 
calculations place the nodal line about 2 km. west of the latter line. 
The western node of the binodal oscillations was not observed by Forel, but he 
places the eastern node as being approximately on the transversal through Morges ; 
the calculated value for this node gives it as the transversal about 2 km. east of the 
transversal through Morges. 
When it is considered that the exact determination of nodes by observation is very 
difficult, and when allowance is made for differences in mean lake level, the results 
that have been obtained by calculation may be regarded as extremely good, and they 
indicate that the method gives a most satisfactory solution of the problem of free 
longitudinal motion in a lake. 
12. Remarks on Methods of Calculation. 
The processes of successive integration require a high degree of nominal accuracy 
throughout the work of calculation ; otherwise the values that are obtained for 
I n (0, 2) will diverge rapidly from the true values as n increases. Whatever be 
the degree of accuracy of p(x), it is essential that there should be as exact a corre- 
spondence as possible between the adopted function and the calculations based on it. 
The difference between theoretical results and observations will then not be due to the 
method of calculation. The experience acquired in the present instance shows that 
when there are considerable irregularities in p(x) a great many values should be 
used in the numerical representation of the function. For values of x between 
0 and 0*3, measured from the west end of the lake, as usual, it would have led to 
better results if considerably smaller intervals in x had been chosen. Values of 
p(x) at intervals of 0*002 in this region would have led to a large increase in 
accuracy. As it is, the values of I„(0, l) are certainly correct to 1 in 10,000, 
but this degree of accuracy is not sufficient for the higher roots of the 
period equation. For the first few values of w a much greater degree of accuracy 
is desirable. 
There is nothing in the theoretical basis for the partition of the lake into western 
and eastern systems (see § 4) that requires the partition at x= 1. It can be at any 
suitable value of x, and in the present case the division at a; = 0'3 would have been 
better, as the value of p(x) is most irregular between £C = 0 and x = 0'3, the remain- 
ing portion being a smooth curve. 
The method of solution of the problem, as expounded in §§ 2—4, is due to Professor 
J. Proudman, and this paper is simply an application of his method. The calcula- 
tions were begun by R. Baldwin, and a graphical method of integration was used : 
his results indicated the need for a much greater degree of accuracy. The work 
