per
denotabimus, idem prodire debebit, sive integratio primo instituatur secundum
ac dein secundum
sive ordine inverso. At habemus indefinite, considerando
tamquam constantem,
![{\displaystyle \int yd\varphi ={\tfrac {tu'-ut'}{r(tt+uu)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15b1b1353da43d51489134dc064dea69eebef095)
uti per differentiationem secundum
![{\displaystyle \varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)
facile confirmatur. Constans non adiicienda, siquidem integrale a
![{\displaystyle \varphi =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/192287b02f5764a18fe39f37b8199d72000aa220)
incipiendum supponamus, quoniam pro
![{\displaystyle \varphi =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/192287b02f5764a18fe39f37b8199d72000aa220)
fit
![{\displaystyle {\tfrac {tu'-ut'}{r(tt+uu)}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0270c9fb1bd29dc24f19c8e324dab8a1c190a754)
. Quare quum manifesto
![{\displaystyle {\tfrac {tu'-ut'}{r(tt+uu)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98d53a72c34be55d8aca6728d0eb7d76ddbe445f)
etiam evanescat pro
![{\displaystyle \varphi =360^{o},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb31b23083ce81e3ea5f46c341f3982bcb0010eb)
integrale
![{\displaystyle \int yd\varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f34088f897f1966d83b7ba01b0bbb25aacb6c0e6)
a
![{\displaystyle \varphi =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/192287b02f5764a18fe39f37b8199d72000aa220)
usque ad
![{\displaystyle \varphi =360^{o}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d58a21075afa4235401640b7cae32f28ed02bd9)
fit
![{\displaystyle =0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1fc73f4650c89ab3162da3193b1a57e041977b23)
manente
![{\displaystyle r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
indefinita. Hinc autem sequitur
Perinde habemus indefinite, considerando
tamquam constantem,
![{\displaystyle \int ydr={\frac {tt'+uu'}{tt+uu}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d1f1119acde7f4d8d0352a11dab9c94d8871ffa)
uti aeque facile per differentiationem secundum
![{\displaystyle r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
confirmatur; hic quoque constans non adiicienda, integrali ab
![{\displaystyle r=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/894a83e863728b4ee2e12f3a999a09f5f2bf1c89)
incipiente. Quapropter integrale ab
![{\displaystyle r=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/894a83e863728b4ee2e12f3a999a09f5f2bf1c89)
usque ad
![{\displaystyle r=R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a088bb7b6b4be9ff6b0fdb33fc1dd53af91e356)
extensum fit per ea, quae in art. praec. demonstrata sunt,
![{\displaystyle ={\frac {TT'+UU'}{TT+UU}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aca738fd905bf0e6eb59b99e0efd62229acf0dd1)
adeoque per theorema art. praec. semper quantitas positiva pro quolibet valore reali ipsius
![{\displaystyle \varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e)
. Hinc etiam
![{\displaystyle \Omega }](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
, i. e. valor integralis
![{\displaystyle \int {\tfrac {TT'+UU'}{TT+UU}}d\varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f68deaf0057109c82afe75ffc5190ba273776ae3)
a
![{\displaystyle \varphi =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/192287b02f5764a18fe39f37b8199d72000aa220)
usque ad
![{\displaystyle \varphi =360^{o},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb31b23083ce81e3ea5f46c341f3982bcb0010eb)
necessario fit quantitas positiva
[1]. Quod est absurdum, quoniam eandem quantitatem antea invenimus
![{\displaystyle =0:}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df6233bd905327147f5cfb1f687e87c0a86de432)
suppositio itaque consistere nequit, theorematisque veritas hinc evicta est.
3.
Functio
per substitutionem
transit in
nec non per substitutionem
in
Quodsi igitur pro valoribus determinatis ipsarum
puta pro
simul provenit
(quales valores exstare in art. praec. demonstratum est),
per utramque substitutionem
- ↑ Uti iam per se manifestum est. Ceterum integrale indefinitum facile eruitur
, atque aliunde demonstrari potest (per se enim nondum obvium est, quemnam valorem ex infinite multis functioni multiformi
competentibus pro
adoptare oporteat), huius valorem usque ad
extensum statui debere
sive
. Sed hoc ad institutum nostrum non est necessarium.