136 METHODS OF ABBREVIATION. [CHAP. IX. 



Next, we will eliminate/, /?, and g from the original system 

 of equations, and then determine v in relation to h, d, and m. 

 We will in this case eliminate p and / together. On addition of 

 (1),(2), and (3), we get 



v{\-p(\-f)(\-h)-f(\-p)(\-h)-h(\-p)(l-f)} 



+ pd+fd = 0. 



Developing this with reference to p and /, we have 

 (v -f Zd)pf + (vh + d)p(l -/) + ( vh + (1 - P)f 



Whence the result of elimination will be 



(v + 2<f) (vh + d) (vh + d)v(l - h) = 0. 



Now v + 2d=v + d+d, which by Prop. I. is reducible to v + d. 

 The product of this and the second factor is 



(v + d) (vh + d), 

 which by Prop. II. reduces to 



d + v (vh) or vh -f d. 



In like manner, this result, multiplied by the third factor, gives 

 simply vh + d. Lastly, this multiplied by the fourth factor, 

 v (1 - A), gives, as the final equation, 



vd(l-h) = Q. (8) 



It remains to eliminate g from (5) and (6). The result is 



v (I - m) = 0. (9) 



Finally, the equations (4), (8), and (9) give on addition 

 v(l-d) + vd(l-h) + v(l-m) = 0, 



from which we have 



_ _ _ 



~ 1-d + d(l -h) +1-TO* 



And the development of this result gives 



v = - hdm, 

 f which the interpretation is, Virtue is a habit accompanied by 



