1820.] Mr. Adams's Mathematical Problem. 419 



V 



^,— 1 _ A . ?)"•-' (I.) 



i;"'-" = B . v"-" (II.) 



t;"-' = C . ^^"-"^ (III.) 



^'=!r§ •"■"■= (' + (")'+("r+(r+^-) 



t;'-' = D . z)-^-' , (IV.) 



&c. &c. 

 Hence we obtain 



II v-w B '^ ^ 



m= T~=^ = B~c •^'~' ^^^-^ 



^ = ^=s^-"- (TO) 



&c 



The equation marked V. when n = -, gives 

 r r •+-+I-1 +(~) +to»» terms ,„ 



The same as given by Mr. Landon, p. 6, Residual Analysis. 

 If in equation I. we write - for w, we shall have 



n 



The same in substance is given in the Edinburgh Philosophical 

 Journal for July, 1820. 



Now it is evident that each of the foregoing series was produced 

 from their corresponding rootn ; and it is also evident that those 



2d2 



