202 L. EULERI OPERA POSTHUMA. Arithmetica. 



"" Demonstbatio. Sumto emm a;==^ el y=^ prodit ^a7-i-7yy==2. Nofttm autem est omnes pole^ 

 fftates formulae xx-^lyy in eJidem formula contineri, quandoquidem est '•'■•■'•">'•! 



'^ -' ■' '■'"' ' • (aa-i-7W)(cc-H7rfrf) = (ac=b76rf)'H-7(arfi^'fe)*. 



muioirp .HT^mmt jiodoii{t * . ■ -■ iiiir^on- i ii 



Hinc iffitur per factores imaginarios jerit . J > . 



. "=• . '^ " ;'|) !o.''. !>oup ^fninolKi!)»;.: ;» eriiim.' 



-, lH_7 lH->/-7 l-y-7 n« /t.-f-'/-7\" /1-y-7\" 



o»p..i,. 2 = V=Lr__!.____I, entergo 2^'= (^-.-^) . (— ^) . 



A^V?^\ . aut^W , . ,."!^;^m. ,Jfitp^tate«. s^sqii^nti modo progrediuntur 



■ .•sninir.fi i\'. ..fii>i(! nflrm £<♦ OTMj-)obmq' nflf'— .S-Hy— 7 /\-^V —l^ ..<. »i 



vf» "Tia 1— »i8 'jenni ^ \ 2 / ^ uuiup 



_5-t/-7 l-f-y-7 Y 



2 ~^ 2 y 



.i>i 



|_3T/_7 



=(-■^=-0* 



2 



' , etc. etc. 



lm»«. ?«a »ool' 



Harum formularum ambae partes seriem recurrentem constituunt, cujus scala relationis est 1, —2, unde si 



ex. gr. ponatur "*" ~ = A, et quia omnes hae formulae per 2 diitriduntur, islae formulae sequenti modo 



continuantur: 



2A = i-\-V-l , , 2^8=-31-3V-7 



^Zt XfX »nm 2^*=-3-hV-7.,^ -.. 2^«=:-5-17V-7 



^A^^-^—V-l ^^^•'^S^-ll V— 7 



2A^=l-3V-7 2A"=67-h23V-7 



2^^=ll-V-7 2^^^^= — 47 H- i5 V— 7 



2-4«= 9 -4- 5 V— 7 2yli3= — 181 — V— 7 



U\ I ^V^T 'rr.njHM 2^7_,__|3_^7y__7 gtc. y 



Cum igitur sit 



ergo 2'^=18Ph-7. 

 Ratio autem scalae relationis in hoc sita est, quod si ponalur 



oujv uiUiia 'uif. ■* — 2 ''f «Wm -^ 2 — 2 



et sumlis quadratis erit zz=zz — 2, unde nascilur scala relationis 1, —2, In superiori pro^essione, ubi omne« 

 termini in forma a-i-fcV — 7 continentur, ii casus maxime sunt notatu digni, iquibus h est vel -+-1, vel — 1, 

 quibus casibus pars rationalis fit maxima. Hincque sequens problema omninO peculiarem postulat solutionem. 



Prorlema. Cum sit uli vidimus ( ) = > investigare eos exponentes n, pro quibus fit 



b=dt.\y id quod fieri observavimus casibus n = l, 2, 3, 5, 13. Qiiaeranlur igilur casus sequentes. 



SoLUTio. Cum esse debeat b = ±\, reducalur formula ad hanc formam p(cos9)-i-V— l.sinyi) 



erilque ,s, , . 



pcos9)=— et j)sin93 = — V7, unde fit tang9D=V7, hincque antp—V— et cos9=V g-, sicque erit,^==y5jj 



Invento igitiir angnlo 9, ut «it tang.9) = V7 erit primo ' 



\-^~y — 7 1 , /1-H"/ 7\" 'i j 



vwt 4 t' — .ir^; V2(cos9PH-V— l.siny) ideoque i ) =2 (cosny-HV- Usini 



