CHAP. X.] CONDITIONS OF A PERFECT METHOD. 151 



chapter. There are, however, conditions under which the latter 

 method assumes a more perfect form than it otherwise bears. To 

 make the one fundamental condition expressed by the equation 



x(l -x) = 0, 



the universal type of form, would give a unity of character to 

 both processes and results, which would not else be attainable. 

 Were brevity or convenience the only valuable quality of a me- 

 thod, no advantage would flow from the adoption of such a prin- 

 ciple. For to impose upon every step of a solution the character 

 above described, would involve in some instances no slight la- 

 bour of preliminary reduction. But it is still interesting to know 

 that this can be done, and it is even of some importance to be 

 acquainted with the conditions under which such a form of solu- 

 tion would spontaneously present itself. Some of these points 

 will be considered in-the present chapter. 



PROPOSITION I. 



3. To reduce any equation among logical symbols to the form 

 V= 0, in which V satisfies the law of duality, 



V(\ - V) = 0. 



It is shown in Chap. v. Prop. 4, that the above condition is 

 satisfied whenever V is the sum of a series of constituents. And 

 it is evident from Prop. 2, Chap. vi. that all equations are equi- 

 valent which, when reduced by transposition to the form V = 0, 

 produce, by development of the first member, the same series of 

 constituents with coefficients which do not vanish ; the particular 

 numerical values of those coefficients being immaterial. 



Hence the object of this Proposition may always be accom- 

 plished by bringing all the terms of an equation to the first side, 

 fully expanding that member, and changing in the result all the co- 

 efficients which do not vanish into unity, except such as have already 

 that value. 



But as the development of functions containing many sym- 

 bols conducts us to expressions inconvenient from their great 



