636 MESSRS A. T. DOODSON, R. M. CAREY, AND R. BALDWIN, THEORETICAL 
by the omission of the subsequent terms, but the relative importance of the various 
terms, and the accuracy of the solution, can be tested afterwards. Since the most 
important terms are those for which n is small, it is best to deal with the series 
where 
+ . . . + (-l)'J, = 0 . . ' (27) 
j*=l/(*A) O 28 ) 
The result of Horner’s method is to give the first three roots as follows 
^ = 5-2764, k\= -18952 \ 
^ 2 = 1-1725, k\ 2 = -8529 l (29) 
-7459, k\ 3 = 1-3407 1 
We have now to estimate the accuracy of these results. In the first place, the 
accuracy of the work, regarding J 0 , Ji, . . . as absolutely accurate, may be tested 
by substituting the above values of kX on the right-hand side of the equation 
(/c\)J 1 = J 0 + ( k A.) 2 J 2 -(kX) 3 J 3 + 
whence division by J\ should give the value of ; the effect of terms beyond J,. is 
easily estimated by this process, and it was shown that the values of k\ as given above 
were not affected, to the order given, by taking terms beyond J 13 into account. But 
the J’s are only known to a limited degree of accuracy, and this is not great enough 
for the third decimal of to be considered as more than approximately correct, and 
the second decimal of >c\ 3 is only an approximation. The value of k\ 1} however, is 
probably correct to at least four decimals. 
The actual periods are given by the relation 
T = 2tt /(g\,y, ..... (30) 
where g, expressed in lake units, has the value 9807 divided by 17052 x 10 6 . 
Introducing the factor k once for all, we have 
T = 2TTK i ff~ h /i s i =--- 32-417/^s minutes .... (31) 
Therefore, the values of the first three periods are respectively 
T x = 74'45 minutes, -i 
T 2 = 35T „ i . (32) 
T 3 = 28 „ J 
and all the figures here given may be regarded as significant. 
10. Determination of the Elevation £ and of the Nodes. 
The elevation £ may be calculated by the formula 
£ = - j^(- *), 
and this function is given in Table VIII at intervals of 07 in x. The value of x is 
measured from the western end. Application of the method discussed in § 4 was 
