824 ON THE PERIODS AND NODES OF LOCHS EARN AND TREIG. 
simple a mathematical approximation as possible. There is here, of course, an element 
of arbitrariness ; but we may expect from the general conclusions of the Hydrodynamic 
Theory that, within reasonable limits, the results will not be greatly affected by the 
particular form of approximation selected. 
The two parts of the normal curve are now represented by 
o=h(1-v?/a’?), o=h(1 -v?/a?) = ‘ : c : (iy 
say 
Gali, 4 ; * ; : 3 é : (2), 
where «=1-—v’/a” or 1-—v"/a? according as the section in question is towards the 
deeper or the shallower end. The depth of the common vertex of the two parabolas is 
still at our disposal, and its choice is another arbitrary element in our calculations. 
To avoid any possibility of bias, we elected to determine / so that the sum of the squares 
of the differences between the observed and calculated values of « at the sounding lines 
should be a minimum. That is, « denoting the observed value, given, for example, in 
the column of the table on p. 826, we choose h so that 2(ha—c)? is a minimum. We 
thus get 
b= SouSe 7). - city a ee gn (3), 
The biparabolic curve which is to replace the normal curve of the lake is now known. 
The divergence of the actual curve from the theoretical one will be seen from figures (1) 
and (2), where the dots indicate the points on the actual normal curve at the various 
sounding lines. Theoretically it would be necessary to correct for these deviations by 
RayLeEiIcH’s method (H.T.S. § 21); but in practice that has not, as yet, been found 
necessary. 
§ 4. The rest of the work in the case of each’lake consists in solving by approxima- 
tion the equations (44) and (45) for the Periods; and equations furnished by (46) and 
(47), or by (65), for the Nodes. 
The tables on pp. 826, 839 give the principal elements in the calculation of h for the 
two lakcs. To secure accuracy, the logarithms of the various quantities were tabulated 
in the unabridged scheme ; so that any gross error in the work might be readily detected 
by watching the progression of the differences, 
For convenience of reference, we have also inserted in the tables the co-ordinates 
of the various nodes, which are calculated later on in this paper. 
