84 Mb DE morgan ON THE GENERAL EQUATION OF 



We also find from (33), for substitution in (21), 

 -:-:-:: l^ — l^A + loA' : m^^ — m^A + m^A^ : n^, — n,A + n^A^ (35). 



The equation (34) must have all its roots possible. For from (31) 

 it appears that A' and A" cannot be of the forms X + m V-l and X— m \/- 1, 

 unless a' and a", /3' and /3", 7' and 7" are of the same form ; from which, 

 since 



{K + xV'^){o-(p\/'^) - («-x\/^)(0 + 0\/^T) 



is of the form k\/ — 1, it will follow that p, q, and r (4) must be of 

 this form : which is inconsistent with (32), if we suppose V^ positive ; 

 since it may be seen from (31), and will presently appear otherwise, 

 that a is possible when A is possible. 



We might find equations of the third degree to determine jh q, &c. 

 but it will be more convenient to express them in terms of A, &c., 

 supposed to be found from (34). To do this, let a,,, a^, I,,, I,, &c. (5) 

 and (6), be found in terms of A, a, he. by substituting the values of 

 a, h, a, h, he. from (31). The results, after reduction, are 



a,,=A'A"jf +A"Ap" +AA'p"', a,=U'+'^")f +{A"+A)p" +{A+A')p"% 

 h,^A'A"(f +A"Aq" +AA'q"\ b={A'+A")q' +{A"+A)q" +{A+A')q"% 

 c„=A'A"f^ +A"Ar" +AA'r"', c=U'+^"V +U"+A)r" +{A+A')r"\ 



..(36), 



l„=A'A"qr+A"Aqr'+AA'q"r", 1,={A'+A")qr+{A"+A)q'r'+{A+A')q"r", 



m„=A'A"rp+A"Ar'p+AAy'p", m={A'+A")rp+{A"+A)rp'+{A+A')r"p", 



n,=A'A"pq+A"Ap'q'+AA'p"q", n=U'+^")Pq+i^"+-^)p'q'+i^+^')P'Y' 



which equations, with those marked (17), give the following values 

 of p'', qr, he. 



a,-a^A + a,A' _ l„ - l,A + kA' 



^~ {A-A'){A-A")' ^ {A-A'){A-A"y 



" — ^i i~^t ^ +hoA ^ _ m,,-m,A + irigA^ . 



^'~ {A-A%A-A"y ''P~ {A-A'){A-A") ^^^' 



•i _ C// — g, ^ +CoA ^ _ n,, - n,A + n^A' 



^ ~ XA^^t^ - ^"') ' ^^~ {A- A) {A - A") ■ 



