154 CONDITIONS OF A PERFECT METHOD. [CHAP. X. 



5. A system of propositions may be termed independent, 

 when it is not possible to deduce from any portion of the system 

 a conclusion deducible from any other portion of it. Supposing 

 the equations representing those propositions all reduced to the 



form 



7-0, 



then the above condition implies that no constituent which can 

 be made to appear in the development of a particular function V 

 of the system, can be made to appear in the development of any 

 other function V of the same system. When this condition is 

 not satisfied, the equations of the system are not independent. 

 This may happen in various cases. Let all the equations satisfy 

 in their first members the law of duality, then if there appears a 

 positive term x in the expansion of one equation, and a term xy 

 in that of another, the equations are not independent, for the 

 term x is further developable into xy + x ( 1 - y), and the equation 



is thus involved in both the equations of the system. Again, let 

 a term xy appear in one equation, and a term xz in another. 

 Both these may be developed so as to give the common consti- 

 tuent xyz. And other cases may easily be imagined in which 

 premises which appear at first sight to be quite independent are 

 not really so. Whenever equations of the form V= are thus 

 not truly independent, though individually they may satisfy the 

 law of duality, 



7(1 - 7) = 0, 



the equivalent equation obtained by adding them together will 

 not satisfy that condition, unless sufficient reductions by the me- 

 thod of the present chapter have been performed. When, on 

 the other hand, the equations of a system both satisfy the above 

 law, and are independent of each other, their sum will also sa- 

 tisfy the same law. I have dwelt upon these points at greater 

 length than would otherwise have been necessary, because it ap- 

 pears to me to be important to endeavour to form to ourselves, 

 and to keep before us in all our investigations, the pattern of an 

 ideal perfection, the object and the guide of future efforts. In 



