( 367 ) 
2m? +1 2n? +1 m? —1 Sm 
Ee TA TETE Cy a +, eater Cp 4+ -—— ae + 
m(m?+2) — n(n?-+-2) m(m? +2) m?+2 
on n?—] 
lem eet eee tp Dey MI Ta ea, badly, oe ROL AG, en 
nr? +2 n(n? +2) 
For the first interval considered here the first of the equations 
(A) is g,—9,+hn=0. This equation may also be written in the general 
form by putting — g, + 29, — 9, + him = 0, thus assuming that the 
value of y preceding g, and c belonging to the first observation are 
both equal to g,. In the same way c belonging to the last observation 
is equal to the last g of the last interval. Between each three 
consecutive quantities c, therefore, a relation exists of the form (2) 
and two other equations are added to the beginning and to the end 
of this series, each containing only two values ¢ derived from the 
formulae for g, and gy. Let cq and cy be the first two and Cy and 
c- the last two quantities c, then we obtain by substituting ¢, for 
Gn Cp and cy for cy the first condition, and by substituting c for 
Jy =Cq and ce, for c, the last condition of the series which determine 
the values c. 
If the lengths of the ince intervals are represented by u and 
y these equations are: 
(2u*+-1) ca = + Qu — (u—1) 
(2y7-- 1) ¢. = + 3n?Q, — (P°—1) oe, 
The series (4) and these two equations determine all the quantities 
c. If we solve them by approximation our purpose is soon gained ; 
n Qin +m Qn 
we assume to the first approximation cg = ————— and ¢, and cz 
m+n 
equal to the values of Q of the first and the last interval respectively. 
From the equations (B) we derive the first corrections A, c,, A, cj, 
ete. and A, ¢g is derived from the formula: 
| 2m 41 2n? +1 m*—l n-—l 
mm? +2) ' nn" +2) ag | 
fee — A, « — ———- A, er. 
eg FL (no) ea a cope Oe | 
In this interpolation we determine g; and $; of an interval of m 
units according to the formulae: 
1 eq ml Cp—=C 1 
n= (Gale tot ')— Ee kn + 5 ty + > Am ë 
2m 2 
1 . Wi ja l , 
8=5+(4-%&)i-(F ith se), Rr ke 
2. Im the previous section the observed and the interpolated 
quantities ‚S, occurring in the problem discussed, form a series of 
