Schubei'Vs investigation of R'ejjler's Prohleiu. 



9 



Supposing now h = o, consequently h, =^ x^ we sliall obtain 



(2) -4. (//) -u+ XM 1 + 



X 



172:"^ 



+.• 



X 



n 



X • ^^ * # « re 



n 



n J 



the letters it, Mj, tt^, u„, denoting tlie value of the function -4/ (?/). 



and its differentials or fluxions, 



lar case when x 





V-^,in the pavlicn- 



0. 



^ 6. The diflPerentials of the functions of ?/, 9, ^{', are necessarily' 



F 



of this form : 



f'. A]j, d4/=4'', d^, df = f". d?/, d-J/'='4/". d?/. 

 and so on ; therefore, since ^ is a function of x, (§ 4*), we obtain 





d-j^ , dy 

 dy d.r 



V. 



Ay 



ii,V 



, etc. A continual differentiation will there- 



fore give us the following equations : 





^'■% ■> 



dd^ 



da; 2 



(3)<: 



f'(^)V^^"^ 



da'2 ' 



d»,^ 

 do?"' 



^"Hdx)'+S^"- d}c-da> 



«5i/ ddi/ d^!/ 



Ld 



X 



•\dx 



^J\ 



+ 



6 r . ( 



d.vV dd^ 



/^i^^ 



dif-d-F'+HHd-;^j^+H". 



d^ diy , d^^ 







Moreover, a continual differentiation of the equation 



y +x,^ 



gives 



(^l^x.(p')f\y + <p.dx; o = (l + x. <?') My + x. o". dy* +2 <?'. da:d?/ ; 

 o = (l-i-x,<p') H^y+Sx . ^". dj/dd^ +a: . <?«'".dy*+S ^'. da^ddi/-f3 4)", did?/ =, etc. 



or since x 



(§ 5), 



= d^ + ^ . da: ; o=dd^+2 <?>'. dxdy ; o — d^y+5p\ dxddi/+3 <p", d.rdi/»; etc. 



f ; which value being substituted 



the first of which arives V- 



* do- 



in the following equation, and so on, we shall obtdin _ 



(4) 



d^ 



d-r 



d*y 

 da* 



^» d'F"^ + ® ^-^ » di» 



6f.(?T-3(9).y; 



+ S4 ? . (?')'+ 36 (f )% p. / +4 (p)\ 9"; etc. 



2 



? 



