94 Prof. Boole on the conditions by which the Solutions of 



Now the quantities X, fi, v, &c. are individually J 0. More- 

 over, they are subject to no other relations than the above. Our 

 object, then, is to seek the relations among w, Cj, Cg, CjjOj, Cj/?^^ 

 which are necessary in order that the above conditions may be 

 satisfied. 



For this purpose we must, and the rule is of general applica- 

 tion, determine as many of the quantities X, /a, v, &c. as we can 

 in terms of «;, Cj, c^, &c., and make their expressions J 0. These 

 ■will furnish a part of the conditions sought. We must substitute 

 the above expressions in the equations of the system (3) (4) which 

 remain, and, supposing those residual equations to be n in num- 

 ber, find from them the expressions of n more of the quantities 

 X, /x, V in terms of the quantities which remain, and of the known 

 quantities Wy Cj, c^ &c. We must make these expressions also 

 ^0, and from the inequations thus formed eliminate by the 

 previous proposition such of the positive quantities X, fi, v as are 

 still left. This will furnish the remaining conditions among the 

 constants Wy c„ Cg, &c. In the steps of this process we shall 

 have successively introduced all the conditions X J 0, /x J . . . 

 i; J 0, and shall therefore have obtained all the equations con- 

 necting the elements w, c„ Cg, c^p^ and c^jOg- 



Thus, from the third, fourth, and fifth equations of (3), we find 



<t=w—CyP^ v=w—c^P2 X=Ci/>, + C2J92— w, 

 furnishing the conditions 



w— CjjOi^O «^— CgjOg^O Cip^—c^p^—w^O. (5) 



Substituting the values of o-, v, and X in the remaining equa- 

 tions of (3) and (4), we find on transposition, 



fl + T^CciO—p^) 

 fl+p-\-T-\-V=il—W. 



Hence, selecting p, t, and v as the quantities to be determined, 

 we have 



p=Ci(l-;?i)-/A 



v=l^w-Ci{l'-py)-c^{l-p^)-hfi; 

 whence, therefore, 



\-w^c^{\-p^)-c^[\-p^-\-ti = 0. 

 Or, in order to eliminate fi, , 



