tendum imprimis exponentes, si positivi, gradus differentiationis, \sin\psi negativi, gradus integrationis denotare ; \sin\psi autem aequales nihilo, tunc argumentum esse, quantitatem illam, cui huju\sinodi additur exponens neque differentiatione, neque integratione opus habere, sed potius uti est, relinquendam ; verum hæc omnia clarius exemplis aliquotperspici posse existimo. Habendum sit itaque differentiale
ipsius
facto
series hunc indicat valorem
seu
![{\displaystyle y\,dx+x\,dy\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/659fc4c9bcf0cc6e1a6b9db62d616587c4b345e5)
si
series fiet
![{\displaystyle x^{2}y^{0}+2x^{1}y^{1}+x^{0}y^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee318d4841d8ce35a95eabb20490c64d4abbadf2)
unde obtinebitur differentiale ![{\displaystyle 2^{\mathrm {dum} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/513d9ab080dbf8d484671049794d6fbdc4e2ee0d)
![{\displaystyle y\,d^{2}x+2dy\,dx+x\,d^{2}y\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3248bee729e4f1f9f1f04ec27d9a23139cf114c)
eodem modo, si
fiet differentiale ![{\displaystyle 3^{\mathrm {tium} }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c61b0ec3d6243227d48a968442f509e7bcb8b711)
![{\displaystyle y\,d^{3}x+3dy\,d^{2}x+3d^{2}y\,dx+x\,d^{3}y\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff6121de0e91db491e7ea785ce9b2844fe15eb24)
existente nempe etiam
fluente ; atque idem dicitur de cæteris differentiationis gradibus. Veniamus nunc ad integrationes. Quæratur integrale hujus quantitatis
substituto itaque in série
loco
et facto
(quoniam intégrale quod quæritur est
) in hanc ipsam transformabitur
![{\displaystyle dx^{-1}y^{0}-dx^{-2}y^{1}+dx^{-3}y^{2}-dx^{-4}y^{3}+dx^{-3}y^{4}-dx^{-6}y^{5}+\ldots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2564dfe0a70f48c5fc0b465283262b96da03e169)
Porro
integrale
seu integrale
ipsius
et generatim
![{\displaystyle dx^{-m}={\frac {x^{m}}{2.3.4\ldots m\,dx^{m-1}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ca1ec5a2de31bd15d0e43ae34f0a9bb9673395c)
posito nempe semper
constanti ; hoc enim per harum quantitatum differentiationem videre est, namque
![{\displaystyle d{\frac {x^{2}}{2dx}}=x,\qquad {\frac {x^{3}}{2.3dx^{2}}}={\frac {x^{2}}{2dx}},\qquad \ldots \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd4792081b657dd8ec62c9bff8b949046b9ad14)