FVNCTIONFM. 197 



Solutio. 



Hoc problema in nullo cafuum ha&enus tradla- 

 torum continetur ; verumtamen idonea transformatione 

 ad cafum facillimum reduci poteft. Ponatur enim 

 P P -+ Q.Q.— x x -\-yy — tt y fitque angulis duobus inde- 

 finitis <p et introducendis : 



P~*fin.(J); Qrrfcof (J>; x~ttoJ et jyzzfcof.O 

 ob dxz^dtCm. + ?</ecof.O, et dy— dtcoM-tdMm.Q, erit : 

 ^V-^/(fin.Cpfini+cof.(p)cofd)-^(cof.(t)fin.0-fin.Cf)cof.O) 

 feu dV~tdtco(.(9-<p)-ttdMm.{Q-<p). 

 At eftftdt cof. (0-$)= £« cof. (O-(J>)+i/0 (</-0(J>) fin.(0-(J>) 

 vnde fit ; 



V=|**cof (Q-$)~lftt{dt-{-d$)Cin-(&~<P). 

 Cum igitur haec formula integrabilis efle debeat , ne* 

 ceffe eft, vt fit //Gn.(0-(p)=rfuna. (0 + $). 

 Quare cum fit ti-zz.xx-^yy et tang. = -, hinc an- 

 gulus $ per x ct y determinabitur , cuius valor fub- 

 ftitutus dabit fun&ionem V, per .r et y expreifam. 

 Sit, vt functiones algebraicas fimpliciores eliciamus , 



#fin.(0-(p) = afin.(0 +-(!>) -r-(3cof.(0-4-$)> eiitc l ue 

 \zz kttcoi (0 - (p) •+- 1 a cof.(0 + <J>) »| (3 fin.(0 + (J>) 



vnde,fi eliminetur Jf, prodit : 



2 Vfin,(0-$)=afm.20 + ^cof.2 0^^^y^. 



At euolutis illis angulis, fit : 



ttxcoi:<P-ttyC\n.(pZ(ixcoL<p+ayCm.<p\-§yco(.<p'PxCin.^ 



ideoque tang. (J) .- t^"jE^t et 



B b 3 fec. 



