294 rilY.-Iol..ic;Y ('I- .MtX'l.KS AND NKRVES. 



table. Substituting tor tin- unknown a:, which may repiv- 

 sent any number, the number 1, then thr table expresses 

 that tin- value of the corresponding?/ is 8. If x=5, thru tin- 

 table expresses that y=10. But when the value of x is 

 intermediate between 4 and 5, e.g. 4 2371, the table does not 

 help us; but by the use of the formula the value of tin- 

 corresponding y may easily be found ; it is = 8-4742. 

 The formula may he reversed, and written thus : 



that is to say, for any given value of y we may calculate the 

 corresponding value of a;. It is exactly the same in the case 

 of tin- similar formula : 



V = 3#, 

 which may also hi: written thus : 



<*< = :!// 



fn tin- case, tin -ret'., re. \\ith each Driven value of x corresponds 

 a certain value of ?/, the latter being three times the value 

 of the funnel 1 . In the two corresponding formulae 



y = ux and x=-y, 

 a 



is a somewhat wider expression to this kind of relation; in 

 this case x and y are again the signs of the two correspond- 

 ing series of numbers, a expresses a definite figure which is to 

 be regarded as unchangeable within each particular case. In 

 our first example f 'l. in our second example ,/ = :',. and 

 similarly 'in any other instance may have any other value. 

 Looking now at the following table : 



1 _' :: I 5 <; etc. 

 1 4 -. hi L'.-I 36 etc. 



we see that any number in the lower series is found by 

 multiplying the corresponding number in tin- upper series by 

 itself, as may be exiuessed in the formula 



y-=.<- ./ i.r // = 



