Pro
sive terminis decem.
Pro
sive terminis undecim.
5.
Quum formula
integrale
ab
usque ad
sive integrale
a
usque ad
exacte quidem exhibeat, quoties
in seriem evoluta potestatem
non transscendit, sed approximate tantum, quoties
ultra progreditur, superest, ut errorem, quem inducunt termini proxime sequentes, assignare doceamus. Designemus generaliter per
differentiam inter valorem verum integralis
a
usque ad
atque valorem ex formula prodeuntem, ita ut sit
![{\displaystyle {\begin{aligned}&k=1-R-R^{\prime }-R^{\prime \prime }-R^{\prime \prime \prime }-{\text{etc.}}-R^{(n)}\\&k^{\prime }={\frac {1}{2}}-{\frac {1}{n}}(R^{\prime }+2R^{\prime \prime }+3R^{\prime \prime \prime }+{\text{etc.}}+nR^{(n)})\\&k^{\prime \prime }={\frac {1}{3}}-{\frac {1}{nn}}(R^{\prime }+4R^{\prime \prime }+9R^{\prime \prime \prime }+{\text{etc.}}+nnR^{(n)})\\&k^{\prime \prime \prime }={\frac {1}{4}}-{\frac {1}{n^{2}}}(R^{\prime }+8R^{\prime \prime }+27R^{\prime \prime \prime }+{\text{etc.}}+n^{3}R^{(n)})\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06845220c5442a4769031666f81f8038bac21423)
etc. Patet igitur, si
![{\textstyle y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db9936ddb2761b76fa640fb275cb5d1fa4d6fa23)
evolvatur in seriem
![{\displaystyle K+K^{\prime }t+K^{\prime \prime }tt+K^{\prime \prime \prime }t^{3}+{\text{etc. }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e37cc57ab00e8235debac58c3483f2d3a6cc9bb)
differentiam inter valorem verum integralis
![{\textstyle \int y\operatorname {d} t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/03bd0874f9e35d7f6a9932836a3642e7ccaa1a57)
atque valorem approximatum formulae exprimi per
![{\displaystyle Kk+K^{\prime }k^{\prime }+K^{\prime \prime }k^{\prime \prime }+K^{\prime \prime \prime }k^{\prime \prime \prime }+{\text{etc.}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c140ff72a74963ff08ea0a4bba7dc352ec7af46c)
Sed manifesto
![{\textstyle k^{\prime \prime }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d6f4dd074a3d73725251c2f084829716d2a9b3b)
etc. usque ad
![{\textstyle k^{(n)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7008ef5dbd7abb79254489ca4dc3834e60e8d52)
sponte fiunt
![{\textstyle =0{:}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49ee927632eeaa75f8215520bc114dd8858ee111)
correctio itaque formulae approximatae èrit
![{\displaystyle K^{(n+1)}{k}^{(n+1)}+{K}^{(n+2)}{k}^{(n+2)}+{K}^{(n+3)}{k}^{(n+3)}+{\text{etc.}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6850fa47027fa2732b79b1aabdd8bf9dac955a05)
Indolem quantitatum
etc. infra accuratius perscrutabimur; hic sufficiat, valores numericos primae aut secundae, pro singulis valoribus ipsius
apposuisse, ut gradus praecisionis, quam formula approximata affert, inde aestimari possit.{{nop}