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INTÉGRALES
![{\displaystyle {\frac {\operatorname {d} x}{\operatorname {d} z}},\quad {\frac {\operatorname {d} ^{2}x}{\operatorname {d} z^{2}}},\quad {\frac {\operatorname {d} ^{3}x}{\operatorname {d} z^{3}}},\ldots {\frac {\operatorname {d} y}{\operatorname {d} z}},\quad {\frac {\operatorname {d} ^{2}y}{\operatorname {d} z^{2}}},\quad {\frac {\operatorname {d} ^{3}y}{\operatorname {d} z^{3}}},\ldots \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbaec45043ee2e12f3c27b37ec5f84fb5dc8797a)
et, si
et
sont d’autres fonctions de ![{\displaystyle z,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/47989a9b66a4ea8a0ec19e8159749fce8a9a8ca8)
![{\displaystyle X',X'',X''',\ldots ,\quad Y',Y'',Y''',\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/80ea74d436f677f8f89734ec68ede811cec37356)
seront pareillement les symboles respectifs de
![{\displaystyle {\frac {\operatorname {d} X}{\operatorname {d} z}},\quad {\frac {\operatorname {d} ^{2}X}{\operatorname {d} z^{2}}},\quad {\frac {\operatorname {d} ^{3}X}{\operatorname {d} z^{3}}},\ldots {\frac {\operatorname {d} Y}{\operatorname {d} z}},\quad {\frac {\operatorname {d} ^{2}Y}{\operatorname {d} z^{2}}},\quad {\frac {\operatorname {d} ^{3}Y}{\operatorname {d} z^{3}}},\ldots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f50063f94568dceda4328b82cf714d9176229ead)
Nous ne recourrons ainsi aux notations du calcul différentiel ordinaire que lorsqu’il s’agira de représenter des coefficiens différentiels partiels. Ainsi
![{\displaystyle \left({\frac {\operatorname {d} V}{\operatorname {d} x}}\right),\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right),\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right),\ldots \left({\frac {\operatorname {d} V}{\operatorname {d} y}}\right),\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right),\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right),\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0ad9d713cb39d2ab341a079a8fc60db0d814acd)
seront les coefficiens différentiels partiels que l’on obtient pour la fonction
en n’y considérant successivement que
![{\displaystyle x,x',x'',\ldots ,\quad y,y',y'',\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fcb56e15d4902ff2670a2f805b1181213fcdd1b2)
comme variables. En conséquence, les expressions
![{\displaystyle \left({\frac {\operatorname {d} V}{\operatorname {d} x}}\right)',\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)',\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)',\ldots \left({\frac {\operatorname {d} V}{\operatorname {d} y}}\right)',\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)',\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)',\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/024cf8e480cdb521a9b197f96298388d4a858c24)
seront la même chose que
![{\displaystyle {\frac {\operatorname {d} \left({\frac {\operatorname {d} V}{\operatorname {d} x}}\right)}{\operatorname {d} z}},{\frac {\operatorname {d} \left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)}{\operatorname {d} z}},{\frac {\operatorname {d} \left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)}{\operatorname {d} z}},\ldots {\frac {\operatorname {d} \left({\frac {\operatorname {d} V}{\operatorname {d} y}}\right)}{\operatorname {d} z}},{\frac {\operatorname {d} \left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)}{\operatorname {d} z}},{\frac {\operatorname {d} \left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)}{\operatorname {d} z}},\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/4201fd9072074c65c68b46f12ef8080672d33822)
Pareillement, les expressions