METHOD OF TRANSFORMING AND RESOLVING EQUATIONS. 299 



A' A" M' M" M;^' a'" _ ,23.) 



A^" A^" nP^' M^v M'V' M^'^ 

 being determined so as to satisfy the three conditions 

 A' = 0, . . (8.) 

 B' = 0, . . (10.) ' 

 C = 0, . ■ . (14.) 

 without resolving any equation higher than the third degree, by 

 a process which may be presented as ^o\\o^^. 



In virtue of the assumption 21 and of the law (5) of the 

 compoSL of the coeffidents A', B', C t is easy to perceive 

 Shose three coefficients are rational andintegra^^^^^ 

 neous functions of the seven quantities AAA M MM M . 

 of the dimensions one, two, and three respectively ; and there- 

 fore that A' and B' may be developed or decomposed into parts 



as follows : , k> ('}A\ 



A' = A'i,o + A'o,i, (^^-^ 



B =:B',.o + BVi+BV, . • • • (25.) 



the symbol A',, or B',, denoting here a rational and integral 



c S r.f A' A" M" M' M" M'", Miv, which is homogeneous 

 function of A , A , A , ivi , m , i*i , , ^ppree i 



^^f 'rt%o Tm^W'V< ^f then^'e arSt detfmin: 

 S^'tr^SU^of"!': A< A^ 'so as to satisfy the two conditions 



A',o = 0, (26.) 



B',.o=0, (270 



and afterwards determine the three ratios of M', M", M'", M^^ 

 so as to satisfy the three other conditions 



A'o.i = 0, 28. 



B\, = 0, (290 



bC = 0, (3O0 



we shall have decomposed the two conditions (8) and (10), 

 namely, ^ ^ ^^ ^, ^ ^^ 



into five others, and shall have satisfied these five by means of 

 the five first ratios of the set (23), namely 



A' A" M' M" H: . . . . (310 



A^" X^" W M^' M^^' 

 without having yet determined the remaining ratio of that set, 

 namely 



