632 MESSRS A. T. DOODSON, R. M. CAREY, AND R. BALDWIN, THEORETICAL 
require I n (0, x ) and I' M (0, x ) in both systems for values of x from 0 to \a , and equation 
(15) yields 
^ o (I n J'r + L'n Jr) = K(0,«) .... ( 19 ) 
where, on the left-hand side, the arguments are (0, \ a ). 
4. Determination of Periods and Elevations. 
The unit of length will be chosen so that the surface of the lake has tw T o units of 
area, and this unit of length will be called a “ lake unit.” Thus we have a = 2, and 
all the functions on the left-hand side of (19) will have the arguments (0, l). 
It is therefore necessary to evaluate the functions I„(0, x), I' n (0, x) from * = 0 
to x= 1 in both systems, and substitution in (19) will give the values of I„(0, 2) for 
the whole lake. These determine the value of A for which R(0, 2, A) is zero, or, by 
(3), for which 
J)(-A)»I re (0,2) = <> (20) 
The elevation £ is then obtained as in equation (10) ; use is made of the values 
of r„(0, x) in both systems, and the two sets of values are combined as follows. 
Reference to (14) shows that when R(0, 2, A s ) is zero we have 
R(#, 2, A s )^-R(0, x, A s ) = RfO, x, X s )^-R(x, 2, A) . . . . (21) 
ox . ox 
whence 
R(0, x, \ S )/R(x, 2, A s ) is independent of x . . . . (22) 
Consequently 
-^-R(0, x, A s )/— R(.r, 2, A s ) is independent of x . . . (23) 
OX OX 
and the values of the elevation in the w-system are a constant multiple of those in 
the e-system. The two systems can thus be fitted together at the point x=l. 
The relation (21) also affords a verification of the value of A g obtained from (20). 
Application to Lake Geneva. 
5. Construction of Normal Curve. 
The chart used by Wedderburn was that of Hornlimann and Delebecque, 
Atlas des lacs frangais, with a horizontal scale of 1/50,000, and with the depths 
stated in metres. He had drawn, across it thirty-one lines as the surface traces 
of transverse sections. The lengths of these lines were measured, and the area of 
the corresponding transverse section calculated. The product of these two quantities 
gave the corresponding value of p(x). The area of the lake from one end up to a 
transverse section was measured by means of a planimeter, thus giving x, and the 
value of a lake unit was found to be u= 17052 x 10 6 centimetres. The original data 
are given in Table I, and from them were deduced Tables II and III, which give 
the values of p(x) at intervals of 0'01 in x. The normal curve for Geneva is 
illustrated in fig. 1, and fig. 2 shows on a larger scale the western portion of the 
