CHAP. VI.] OF INTERPRETATION. 91 



but that the other coefficients are subject to the law in question, 



so that we have 



3 2 = 3 , &c. 



Now multiply each side of the equation (1) by itself. The re- 

 sult will be 



w = af <j + 2 2 4 + &c. (2) 



This is evident from the fact that it must represent the develop- 

 ment of the equation 



w= V\ 



but it may also be proved by actually squaring (1), and observing 

 that we have 



t l Z = t 1 , *2* = *2, ^2 = 0, &C. 



by the properties of constituents. Now subtracting (2) from (1), 

 we have 



(! - af) #! + (# 2 - #2 2 ) 2 = 0. 



Or, i (1 - x) 1 



Multiply the last equation by ti ; then since ti t z = 0, we have 



! (1 - !) ^ = 0, whence ti = 0. 

 In like manner multiplying the same equation by 2 > we have 



3 ( 1 - # 3 ) t?, = 0, whence t z = 0. 



Thus it may be shown generally that any constituent whose 

 coefficient is not subject to the same fundamental law as the sym- 

 bols themselves must be separately equated to 0. The usual 



form under which such coefficients occur is -. This is the alge- 



braic symbol of infinity. Now the nearer any number approaches 

 to infinity (allowing such an expression), the more does it depart 

 from the condition of satisfying the fundamental law above re- 

 ferred to. 



The symbol -, whose interpretation was previously dis- 



cussed, does not necessarily disobey the law we are here consi- 

 dering, for it admits of the numerical values and 1 indifferently. 

 Its actual interpretation, however, as an indefinite class symbol, 

 cannot, I conceive, except upon the ground of analogy, be de- 



