( 921°) 
After adding and ranging we find: 
a \ 
= = Mg= 427n? | [a,*+2a,? + 8a,’+ 4a,’ + 5a,’ +...) (ra : ) -|- 
8 
+ | a,’ Has + 6a,?+10a,?+ IG | | 
| 3 hehe 4 hadj eee 
r 
0) 
hae $ da 0 
+ | da,’-+ 9a,’ + lda,?+ 30a,? + ll — 5 L 
oe A Ja’ 
+ | a,’ +- 5a,” -+-14a,*-+30a,? + Saline 
ei 
The terms 2 and 3 can be combined into one, namely : 
5 ‘er t dda a, da 
[a,? + 3a,? + 6a,* + 10a,? + ...] G alee ee ) 
The infinite series within the square brackets must be integrated. 
We replace a,, a,.-. ete. by their values, a, — da, a, — 2da …. 
ete. and obtain for instance for the first series: 
CO =d Ie 
daj == 2a da,da + 2a? 
Ba = 3G." = 12a,da + I2da° 
4a,? = 4a? — RU RL Valid faites sehen 
5d, Di — 40a,da + 80 da? 
Mam = Ma, — Zmlm-l a, da + m(m-l da? 
The numbers in the vertical ranges, figuring as coefficients of 
a,?, a,da and da? form arithmetical series of respectively first, 
second and third order. 
The general expression for the sum of m terms of an arithmetical 
series of the nt" order is: 
m(m 1) A m(m—1) (m—2) 2 mm) (m2) (m3) 
On == Pil, : x Be eg EY oie 
TP a ee WEN aga 
Maoh). Semen) id 
OH ne eo Sea 
Wherein ¢, = first term of the series, 4,, A,...A,= first terms of 
the series of differences, 7 == the order number of the series. 
After adding the terms of (8) we thus obtain an expression of 
the following form: 
Pa,* — Qa,da + Rda? 
wherein P, Q and R are functions of m, which are easily found 
60* 
