126 Oft the RtsohUion'(f^Equatimt':»/iiii^^^^^m^m '''^^ 



cle^rfen terms, 'ioi that,! b^'tlW notation of p. 114i^6fe^j|?*ttff 

 the Mathematician, (2.) and (3.), respectively, beddttJ^'' ^f*!'**^ 



J^M^^^.'n •• :^^-^' and/MlO) = . i^^^f^ 

 !NQw'i^)iaye s^^^^ at paragraph 6 of that page, hgJ^^Jl^Y 

 ineaiis of a quadratic, which we shall here represep^^^^jj^j^j^ 



C(-io) = o, • • • / ' ^ M.M 



and four base-\ quadratics which may, respectively, be denoted 



(6.) may be reduced to .^^iV^vvis^tftO 



fend also, hbw, by means of a quadratic, -iaoa^A dai^iifJ A 

 6 2 /.^ N _ Q ^ jiom no ikou-iiz itim. 



and a cubic, , ,.x;(,;ino-^^(%) = Oj • ' m ^sam deotli oi i^fefj 

 ^8,^ j^ liil^ypi^t^Jyj reduced to the base cubiv/ ov/i ni onob ?ijw 

 )«a-i9fiib djiv/ noiij/^^s ^ ;^^3 = J (3) — q; '.^'^^^"P,^'^^^^"9/!} 

 whence, by eliminating Zc^ between this last equation and (7.) 

 (which we may now reduce ta/* (2) = 0)* we arrive at a final 

 biquadratic'^V .'^^«'^ ^^^^ffA •><>'^ ^'*,^>^Ar/ ... 



i ■ ^ ' ii -J i tMr> Mil ■ 



lor determining a?, ; and ^"2, &c. can be found by means of 

 the other equations distinguished by letters; and, consequently, 

 i^', A", &c. are known. _ ,^^j .^.^. ..^ .^^.-^ 



2. In this investigation, if we suppose that jj'^o gjjig jj^Bg 



a; = Q ^? + 4/ (x), . ,rj ^aiL O raoi^9.) 

 then our object is ?iot to satisfy (6.) independent^' tf^^^'^i^li 

 to have that quantity at our disposal for the purpose of satis^ 

 fying (7.) t, but to reduce (6.) to a linear form, and then eli- 

 minate between (6.) and (7.)- 



3. In general, after effecting our fundamental reduction 



/«(Z»)=S(n ....... (10.) 



we may group any two of the quantities h'^ together, as in the 

 above instance, and make their sum equal to zero ; or should 

 it in any case be necessary, we may increase the number of 

 disposable quantities, and, possibly, obtain a (transcendental?) 

 result, by grouping two or more terms of the right-hand side 

 of (10.) together, and making their sums equal to arbitrary 



^ '* The 2nd page of No. iii. (July 1844). 



f The word " base " is used to indicate that they degenerate tof liifear 

 equations. %• ■ - jjiiou A t 



J See Phil. Mag.,S. 3. vol, viii. p. 538, and vol. ix. p. ga" ^^^ §"omfi 



