CHAP. X.] CONDITIONS OF A PERFECT METHOD. 157 



Now A and B must satisfy the condition 



Hence A (confining ourselves for the present to this coefficient) 

 will either be or 1, or a constituent, or the sum of a part of the 

 constituents which involve the symbols x and y. If A = it is 

 evident that E = ; if A is a single constituent, or the sum of a 

 part of the constituents involving x and y, E will be 0. For the 

 full development of A, with respect to x and y, will contain terms 

 with vanishing coefficients, and E is the product of all the co- 

 efficients. Hence when A = 1, Eis equal to A, but in other cases 

 E is equal to 0. Similarly, when B = 1, _E"is equal to B, but in 

 other cases E' vanishes. Hence the expression (2) will consist of 

 that part, if any there be, of (1) in which the coefficients A, B 

 are unity. And this reasoning is general. Suppose, for instance, 

 that V involved the sjmbols #, y, z, t, and that it were required 

 to eliminate x and y. Then if the development of V, with re- 

 ference to z and t, were 



zt + xz(l -f) + y(\ -z)t+ (l-z) (!-*) 

 the result sought would be 



this being that portion of the development of which the co- 

 efficients are unity. 



Hence, if from any system of equations we deduce a single 

 equivalent equation V= 0, V satisfying the condition 



7(1 - 7) = 0, 



the ordinary processes of elimination may be entirely dispensed 

 with, and the single process of development made to supply 

 their place. 



8. It may be that there is no practical advantage in the me- 

 thod thus pointed out, but it possesses a theoretical unity and 

 completeness which render it deserving of regard, and I shall ac- 

 cordingly devote a future chapter (XIV.) to its illustration. The 

 progress of applied mathematics has presented other and signal 

 examples of the reduction of systems of problems or equations to 

 the dominion of some central but pervading law. 



