1 66 Algebra. 



Let us now give to .r a series of values, sav the successive in- 

 tegers from I to lo, and in each case compute the corresponding 

 vahie of v. The results may be expressed in the form 

 )' ! 6 8 lo 12 14 16 18 20 22 24 

 A- I I 2 3 4 5 6 7 8 9 10 

 where any number in the lower line is one of the values of .v and 

 the number immediately above it is the corresponding value of j'. 

 If x= 2 the corresponding value of r is 8, 

 and if .r== 10 the corresponding value of v is 24, 

 and if or be considered to increase from 2 to 10 then at the same 

 time y will increase from 8 to 24, or, starting at .v=2, if -r in- 

 crease by 8, y will increase by 16, or if the increase of .r is 8, the 

 corresponding increase of r is 16. 



Still starting at .^"=2, let us increase x by various amounts and 

 determine the corresponding increase of r. 

 The results may be arranged in the form 



increase of J' | 16 14 12 10 8 6 4 2 



increase ofjr|8 7 6 54321 



We might have started with some other value of x than 2 and 



have obtained similar results. In every observ^ed case we see 



that the increase of v is just twice the increase of .v, or in ever>'' 



observed case 



increase of y _ 

 increase of x ' 

 It is easy to see that this is necessarily the case whatever the 

 value of -v with which we start and whatever the amount by 

 which .V is increased, for if x increases by any amount, 2X will 

 increase by just twice that amount and the change in the value 

 of X does not affect the 4, therefore 2x-\-^, or r, will increase 

 twice as much as .r increases, or 



increase of r 



.-- ^-- =2. 



increase 01 v 



6, Notation. In what follows we deal largely with equa- 

 tions formed by putting r equal to a function of r, and as w^e will 

 make extensiv^e use of the increase in the value of .v and the cor- 



