de nota methodo intcgrajidi. 109 



variaLiles x, y, z, e, p, q, r per u et Septem constara.es arbitiarJas a, b, c, ...li 

 integratione ingiessas exprimantiir (§. ii.), nun hae ipsne expressiones ita 

 eiunt comparatae, ut eas earumque diiTerentialia completa, quantilatxljus 

 a, b, ...h, etiam instar variabilknn habitis, in aequationc proposita substi- 

 tuendo, haec in aequationem inter Septem variabiles a, b, c, . . . h transforme- 

 tur. Jam vero ex supra demonstratis hujus aequationis integratio qnatuor 

 ae^wationibus hujus formae absolritur: 



1) F Ca,b,...h)= v/. [f(a.b,.;.b>,f(ij,b, ..h),h] 



a) F (a,b,...h)= ^//■ [f(a,b,...h)] 



S) F (a, b. . . . h; = v^' [f (a, b, . . . h)]. 



4) F (a,b,.". .h)= ^^I^. 



v,Qtfod si deinde a, t, . . . h per''c(cf<!r varfabiles x, y, . .. n expriniantur, hae' 

 aequationes in has abiturae sunt : 



1) F (x, 7, T, t, u, p/q, r) = -P [f (x, . .". r) , f (t, . . . r), f(x, . , ,r)] 



2> F (x,....r)= -<^{f(x f)] 



3) F (x,....r)= 4.' [f(x,...v^)] 



,,,- . 4) F (x.....r) = '^^'CfCx r)} 



qitarmu Sfsteinate integraiio completa aequationis piopasitae inter octo va»' 

 riabiks ei^ibetnr. 



§■ 12. 

 P r o b 1 e m a IX. 

 Aeqnaho'nem dlfierentialem vulgarem iuter novem vnriabilp^'prt' systema 

 quinque aequationum integrare. 



S o 1 u t i o. 

 Sit aequatio proposita inter novem variabiles n, x, y, z, t, p, q, r, s, haec: 

 du =s= Pdx -j- Qdy-I- Rdzi -f- Sdt -f Tdp -\- Hdq 4 >Vdr -f Jds. 

 Considerando unain liaruni quantitatiim, veluti 's, tam^uam constantem, aequa- 



