6.
Functiones praecedentes sunt algebraicae atque transscendentes a logarithmis circuloque pendentes. Neutiquam vero harum caussa disquisitionem nostram generalem suscipimus, sed potius in gratiam theoriae functionum transscendentium altiorum promovendae, quarum genus amplissimum series nostra complectitur. Huc, inter infinita alia, pertinent coëfficientes ex evolutione functionis
in seriem secundum cosinus angulorum
etc. progredientem orti, de quibus in specie alia occasione fusius agemus. Ad formam seriei nostrae autem illi coëfficientes pluribus modis reduci possunt. Scilicet statuendo
![{\displaystyle (aa+bb-2ab\cos \varphi )^{-n}=\Omega =A+2A^{\prime }\cos \varphi +2A^{\prime \prime }\cos 2\varphi +2A^{\prime \prime \prime }\cos 3\varphi +}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bb10037559115af6cf143946c765419c297e9ec)
etc.
habemus primo
![{\displaystyle {\begin{alignedat}{2}&A&&=a^{-2n}F(n,n,1,{\tfrac {bb}{aa}})\\&A^{\prime }&&=na^{-2n-1}bF(n,n+1,2,{\tfrac {bb}{aa}})\\&A^{\prime \prime }&&={\tfrac {n(n+1)}{1.2}}a^{-2n-2}bbF(n,n+2,3,{\tfrac {bb}{aa}})\\&A^{\prime \prime \prime }&&={\tfrac {n(n+1)(n+2)}{1.2.3}}a^{-2n-3}b^{3}F(n,n+3,4,{\tfrac {bb}{aa}})\\\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58489d18307ef0750c2a6ea7106489a7a0eb0aad)
etc.
Si enim
consideratur tamquam productum ex
in
(designante
quantitatem
, fit
aequalis producto
Quod productum quum identicum esse debeat cum
![{\displaystyle A+A^{\prime }(r+r^{-1})+A^{\prime \prime }(rr+r^{-2})+A^{\prime \prime \prime }(r^{3}+r^{-3})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eced11d351fe94d3fc9e6f8973ab1d98f31ca0a3)
valores supra dati sponte prodeunt.
Porro habemus secundo