( 370 ) 
For the total integral S /, we can also derive a simple form by 
integrating partially: 
z/d°SN? dS d°S "=d 2S 
NE — —— | — ae ih 
ij dt? dt dt? | dt dt? 
a a a 
Ne 
d S 
For the first moment a and the last moment z, ~~ —0, as follows 
dt 
from the first and the last equations belonging to (C). For each 
PS 
interval between two observations a is a constant quantity. Hence 
dt 
we find: 
L = & 6 & (Sg — S-) 
where the summation extends over all the intervals between the 
observations. We can easily find a simple expression for the differential 
quotient of / according to each of the observed values, which may be 
useful when we want not only to interpolate for an intermediate moment 
but when at the same time we have to determine the most probable 
values of the observed quantities. For then the difficulty presents it- 
self how to find the best method for diminishing the amount of the 
minimum value / by applying corrections to the observations, of 
which corrections the mean value is known. 
In doing so heed must be taken that these corrections, being 
errors of observation, shall satisfy the law which determines their 
probabilities as functions of their magnitudes. 
I have not yet reached a satisfactory solution of this problem. 
The following remarks, however, on this subject seemed important 
enough to be communicated. 
3. Let L,, Lg, L, be the observed quantities, free from errors of 
observation, and jf), fy, fr the errors themselves. 
If we have developed the interpolation by means of the quantities 
L and f separately, we obtain the formulae: 
y= Lg + Got + Cg? + Ent? 
hi= Ja Bet Jr Ag Danie et. 
By means of the summation of these two formulae we get: 
Si= Sg + ogt + eg t° + ent’. 
WAR ED 
dt? dt? 
If we apply a partial integration to i dt, we get: 
a 
~ 
x 
aL df aL df it 
dt? dt ee dt 
a a 
