149
ARITHMÉTIQUES.
Soient les formes
,
,
,
… etc.
Faisons
,
,
, etc.
nommons
,
,…
,
…
,
, etc. les indéterminées des
formes
,
,
, etc. et supposons que
se change
en |
par la substitution |
, |
|
![{\displaystyle y=\gamma 'x'+\delta 'y'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71c7eecc13ab3279b031dd363d297f99455a4fdb) |
|
|
………………… |
, |
… |
![{\displaystyle y=\gamma ''x''+\delta ''y''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49cf0f28b834e40852004add51993bfff213ee1a) |
|
|
………………… |
, |
… |
. |
|
Cela posé, comme
se change en
en faisant
et
,
en
en faisant
…
,
en
en faisant
et
, etc. on trouvera
facilement les équations suivantes :
![{\displaystyle \alpha '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cb0468d39268c4405a9286d2cba77c2e4631fed) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) |
|
![{\displaystyle \beta '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d14001f211d8e272b5c10e45c739d320359c48c8) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle -1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac) |
|
![{\displaystyle \gamma '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e2b9e9e12e56bd62af445be6803ec4843e919a) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf) |
|
![{\displaystyle \delta '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aab2d3fbf71b804a6005c7546c058e15f4ea3685) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
|
![{\displaystyle \alpha ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4651b4e0cb2e6bf7726effc97a1c4ca63c799a67) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle \beta '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d14001f211d8e272b5c10e45c739d320359c48c8) |
… |
![{\displaystyle \beta ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ea133d868709de57c71dcb6f576628d79c9fbe8) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle h''\beta '-\alpha '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa59f7847f030c8ae7db2a1a44486c55d851b03c) |
… |
![{\displaystyle \gamma ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b8bb7728f3ec6d2715921dbd21f68b5be7e22d6) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle \delta '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aab2d3fbf71b804a6005c7546c058e15f4ea3685) |
… |
![{\displaystyle \delta ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1c98b36e888b2dffe5c9a7bd7e29dda444adada) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
|
![{\displaystyle \alpha '''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/118ccafdeb1e780b722fbfcfda7ccc389662434e) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle \beta ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ea133d868709de57c71dcb6f576628d79c9fbe8) |
… |
![{\displaystyle \beta '''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43d8aeca3a0b1a51d26bd9f5ec0ffa26757c6247) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle h'''\beta ''-\alpha ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/919d5dd0bcdbc165f4dec3ef8123b44b78388655) |
… |
![{\displaystyle \gamma '''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e210e56606e31f94140ccc7846a4ac3fc8e3c89) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle \delta ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1c98b36e888b2dffe5c9a7bd7e29dda444adada) |
… |
![{\displaystyle \delta '''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a83df20d091000db81dbb64d31bdb948776ff70) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
|
![{\displaystyle \alpha ^{IV}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f74575957401dab8470f807fea3e46cf3780f853) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle \beta '''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43d8aeca3a0b1a51d26bd9f5ec0ffa26757c6247) |
… |
![{\displaystyle \beta ^{IV}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73c803a66c9715dd677dba9f6ed8c94b6f8d1f97) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle h'''\beta '''-\alpha '''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c30fea9fe429034a0c5360f7adac402e584e7920) |
… |
![{\displaystyle \gamma ^{IV}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f3526686b31b6051bf8eabac9978d5feb90815c) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle \delta '''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a83df20d091000db81dbb64d31bdb948776ff70) |
… |
![{\displaystyle \delta ^{IV}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d3c2d84bcf02d1c0f6dd3af14f8a01be67e7785) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
|
|
|
etc. |
|
|
|
etc. |
|
|
|
etc. |
|
|
|
etc.
|
d’où l’on tire
![{\displaystyle \alpha '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cb0468d39268c4405a9286d2cba77c2e4631fed) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2aae8864a3c1fec9585261791a809ddec1489950) |
|
![{\displaystyle \beta '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d14001f211d8e272b5c10e45c739d320359c48c8) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle -1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/704fb0427140d054dd267925495e78164fee9aac) |
|
![{\displaystyle \gamma '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e2b9e9e12e56bd62af445be6803ec4843e919a) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf) |
|
![{\displaystyle \delta '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aab2d3fbf71b804a6005c7546c058e15f4ea3685) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
|
![{\displaystyle \alpha ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4651b4e0cb2e6bf7726effc97a1c4ca63c799a67) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle \beta '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d14001f211d8e272b5c10e45c739d320359c48c8) |
… |
![{\displaystyle \beta ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ea133d868709de57c71dcb6f576628d79c9fbe8) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle h''\beta '-\alpha '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fa59f7847f030c8ae7db2a1a44486c55d851b03c) |
… |
![{\displaystyle \gamma ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b8bb7728f3ec6d2715921dbd21f68b5be7e22d6) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle \delta '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aab2d3fbf71b804a6005c7546c058e15f4ea3685) |
… |
![{\displaystyle \delta ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1c98b36e888b2dffe5c9a7bd7e29dda444adada) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
|
![{\displaystyle \alpha '''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/118ccafdeb1e780b722fbfcfda7ccc389662434e) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle \beta ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ea133d868709de57c71dcb6f576628d79c9fbe8) |
… |
![{\displaystyle \beta '''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43d8aeca3a0b1a51d26bd9f5ec0ffa26757c6247) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle h'''\beta ''-\alpha ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/919d5dd0bcdbc165f4dec3ef8123b44b78388655) |
… |
![{\displaystyle \gamma '''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e210e56606e31f94140ccc7846a4ac3fc8e3c89) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle \delta ''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e1c98b36e888b2dffe5c9a7bd7e29dda444adada) |
… |
![{\displaystyle \delta '''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a83df20d091000db81dbb64d31bdb948776ff70) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
|
![{\displaystyle \alpha ^{IV}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f74575957401dab8470f807fea3e46cf3780f853) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle \beta '''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43d8aeca3a0b1a51d26bd9f5ec0ffa26757c6247) |
… |
![{\displaystyle \beta ^{IV}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73c803a66c9715dd677dba9f6ed8c94b6f8d1f97) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle h'''\beta '''-\alpha '''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c30fea9fe429034a0c5360f7adac402e584e7920) |
… |
![{\displaystyle \gamma ^{IV}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f3526686b31b6051bf8eabac9978d5feb90815c) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
![{\displaystyle \delta '''}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a83df20d091000db81dbb64d31bdb948776ff70) |
… |
![{\displaystyle \delta ^{IV}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d3c2d84bcf02d1c0f6dd3af14f8a01be67e7785) |
![{\displaystyle =}](https://wikimedia.org/api/rest_v1/media/math/render/svg/505a4ceef454c69dffd23792c84b90f488543743) |
|
|
|
etc. |
|
|
|
etc. |
|
|
|
etc. |
|
|
|
etc.
|
il suit du no 159, et de la formation de ces quantités, que les
différentes transformations sont propres.
Cet algorithme très-simple, et auquel on applique facilement
le calcul, est analogue à celui du no 27[1], auquel même il
- ↑ On aurait, d’après la notation du no 27,
![{\displaystyle \beta ^{(n)}=\pm [-h'',h''',-h^{IV},\ldots \pm h^{(n)}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4633cbd13bf4d64e39ded651f3b900f69bbddf97)
,
où les signes ambigus doivent être
,
;
,
;
,
;
,
, suivant que
est de la forme
,
,
,
,
![{\displaystyle \delta ^{(n)}=\pm [h',-h'',-h''',\dots \pm h^{(n)}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5e331848f2fcd594fbeecd2666f75a3309a7254)
,
où les signes, dans les mêmes cas, doivent être
,
;
,
;
,
;
,
.
Mais le désir d’abréger nous empêche d’insister davantage sur ces formules,
qu’au reste chacun pourra confirmer aisément.