PROFEBSOB FriO:-AY ON IMPOSSIBLE EQUATIONS. 239 



(a.) When 2ac + J is negative, we have R < ma ; hence, 

 in this case, the second value of y is also negative, and 

 equation (1) is impossible. 



(j8.) When 2ac -\- h is positive, we have R > ma ; 

 hence the second value of y is positive, and points to a 

 possible root of equation (1), which may be found as 

 follows : — Substituting — ma -j- R for ?/ in equation (2), 

 we get V (2 ax -|" ^) ^^ — "^ 4" ^» 



.*. 2ax -\- b zn ni'd — 2/wa R -|- n^c^ -[- lac -|- h^ 

 or cc zz rr^a -\' c 1- w R. 



Secoiidlyy Let m and a have contrary signs ; then — ma is 

 positive, and if the upper sign be taken in equation (4), 

 the corresponding value of?/ will be positive. In this case, 

 therefore, one of the roots of equation (1) is always possible, 

 and may be found as above. 



(a.) When 2ac '\- b v& positive, R > »ia; hence the 

 second root of equation (3) is negative, and must be 

 rejected as leading to an impossible value of .». 



(jS.) When 2^ -f- 6 is negative, R < ma; hence the 

 values of y given by the formula (4) are both positive, and 

 equation (1) has two possible roots, which may be found as 

 above, or by the ordinary rule for quadratics. 



It may be observed here, that when the roots of (3) ar« 

 imaginary, the quantity 2ac + 6 is essentially negative; 

 from which we see that when m and a have contrary signs, 

 and the roots of (3) are imaginary, the roots of (1) are both 

 possible in the new sense of this term, although they are 

 both imaginary in the ordinary sense. 



By taking successively m-mly and w rz — 1, in equation 

 (1), we obtain the two equations discussed by the author; 

 so that this discussion of equation (1), appears to embrace 

 all the results at which he had arrived up to the time of 

 reading his paper. 



