( 369 ) 
. 
y ke ; JP: Yo 
St = Sp + mt — ie Ge gg 2 Qn) = al + 
Ip + ea 2Qn A 
2m 
m? 
By substituting in the above formula m—t' for ¢, we obtain for S, a 
formula developed according to the ascending powers of £, the interval 
between the moment for which we interpolate and the moment of the 
next observation. It is simpler to find the same formula by imagining 
the interpolation to be made in the inverse direction, so that the quantities 
g and Q change signs and the indices p and q change places. Hence: 
Sm—t = S= Sj — ggd + = Z (Ip + Iq— 2Qm) — Eee fe le Ip+Iq—2Qn Ë 
2m 
m? 
For S;, to be interpolated in the following interval we use: 
3 edt Ig+Gr—2Qn i 
St = Sq + Jt — De (Ig + Ir — 2Qn) ae Sy el + A ; 
Therefore the formulae on either side of each observation are 
different. If in the latter formula ¢ is negative and —‘? is substituted 
for it, the resulting formula differs from the preceding one only in 
the coefficients of the terms of the 3rd degree. The coefficients of 
the terms of the 2nd degree have become equal by satisfying the 
relation (C). 
Therefore we also obtain the interpolated function if, by starting from 
a value (S,) derived from observation, we represent the values of 
S_, and Sp for the moments between that observation and the next 
preceding one and those between that observation and the next 
following one by the formulae: 
S_t—= Sy — gat + gt? — Emt° and S= Sj + got + et? + ent®. 
Taking this as basis, we find: 
+97-3Qm +294 rt Agg H3Qh-gr 3 Ip 99-2 Qm e __ Ja Ir~2Qn 
EERE ERE ear ae ne a n— : 
q m n m? n° 
z/d?SN* ef . 
The integral if Wee) dt, which becomes a minimum through this 
a 
nterpolation, is equal to the sum of the integrals between two con- 
secutive observations, and each of these integrals can be expressed 
in the coefficients of the interval in the following manner: 
a={(e)* geeen Ee sd OY 
n aL 
4 
or: En rg (0g + eg er + €*,)- 
