( 368 ) 
discrete values corresponding to an arithmetical series of the argu- 
ment: now I will remove the restriction of commensurable arguments 
and will make this mode of interpolation applicable to a continuous 
varying quantity and an arbitrary argument by putting for the 
ratio of that series the infinitely small value df. The condition 
pe 
d?SN* 
of minimum then becomes EE dt = Min: 
( 
a 
The formulae for this continuous interpolation may be derived 
independently, but it is shorter to derive them from the corresponding 
formulae of the discrete interpolation developed above. For the present 
I shall put for the lengths of the intervals between which we have 
dS 
to interpolate m’ and n’, for the derived values ae of the interpo- 
lated function g’, to distinguish them from the letters we have used 
in the former problem. 
! ! 
nL u 
Instead of m and 7 we have : and 7 fOP TE 
dt dt 
substitute the quantities g', dt, g', dt, g',dt, and for Q, and Q,, the 
Cry we must 
Bee SEN she 
quantities ———* dt and “—— dt or Qndt and Qy dt. 
m Nn 
After dividing the relations (B) by df and omitting the infinitely 
small values we have: 
a OEE EE 1 
== es G —_ — — == 1 = Pd — ed “ 
mn! Ja a! m7 Pai ni’ 
from which, after dropping the accents, we get: 
et nQm + MQ,  n(Qm — 9) m(Qn— 9) 
ALT m+n 2(m + 7) 2 (m + 7) 
to which we must add as first and last equations: 
EE 
u—G a 
@ 3 B and ge =& + eg 5 AI 
Ja — Q, + 
6 
For k„ we substitute — (gp +9’, — 2 Qn’) (df)*; for 7 we substitute 
m 
ae ; : 
a if ¢ represents the time between the last preceding observation at 
: 
the moment for which we interpolate. These substitutions in the 
formula for S, yield a formula for S,, which, after the omission 
of infinitely small values and accents, is: 
