A1SID OTHER SUUD EQUATIONS. 217 



It is scarcely necessary to say, that in finding these values 



<Jf V 2 a «~+ &, only positive results have been received. 



Ill . . ,, 



Now, according as & is positive or negative, we shall 



obviously have »/ a^ 4- b > or < a. Hence, (o) + 0) 



11 1 



gives 



X -\- t^2ax -{- b z=z OT 2a^ according as b is positive or 

 1 1 1 1 1 1 



negative. 



In like manner, (/)) — (7) gives 



X — j^2ax •4-b zz2a or 0, according as 6 is positive or 

 1 1 1 1 I 1 ' 



negative. 



Again, (6) + (8)j and (6) — (8), give respectively, 



X 4- V^ooj 4- 6 rz 2a ; 

 I 1 1 1 I 



and X — \/2ax 4- ^ in 0. 

 1 111 



From these results it appears, that (5) is the solution of 



either (a) or (/3,) according as b is positive or negative ; 

 I 1 I 



and that (6) is always a solution of (/S). Hence, also, by 



art. 3, cor. 3, and art. 4, when b is negative, (a) is impos- 



1 1 



sihUf and (j8) has two roots, viz., (5) and (6). 



We have already shown that (1), (P), (3), or (3^) is a 

 rigid symbolical solution of the equation (a); and yet we 



now find that neither (5) nor (6), which were both imme- 

 diately obtained from that solution — by merely substituting 

 for the symbol n its arithmetical value — is necessai-ily a solu- 

 tion of that equation. To the experienced analyst, this 

 seeming incongruity will be no matter of surprise. In 

 Professor Young's " General Principles of Analysis," Part 

 I., art. 8, a somewhat analogous case is elegantly discussed. 

 The principle on which such seeming discrepancies as the 

 one above alluded to may be satisfactorily explained, is 

 there developed with that clearness of illustration and logi- 

 cal precision, for which that profound mathematician is so 



2 F 



