22 JOURNAL OF THE 



A.i" \y 

 middle terms above approach — , — indefinitely and the 



A5 A^ 



right members approach, as their limits, cos IPQ, sin IPQ 

 respectively, as A.r and consequently aj^ and a.^ approach 

 zero indefinitely without ever reaching it. Hence taking 



A.t' dx A J dy 



limits and designating lim. — by — and lim. — by — we 



b.s ds 

 have, 



^x dx 



(6) 



• • (7). 



These equations are satisfied by representing ds by the 

 hypotenuse of the right triangle of which dx and dy are 

 the other two sides. Thus if PQ = dx and QI = dy^ then 

 PI = ds; but if PR = dx and RT = dy, then PT = ds. 



In any case, we have the fundamental relation, 

 ds'' = dx'' ^ dy' (8). 



It is usual to represent the derivative of y^(^') with re- 

 spect to X hy f^ {x). 



/^y dy 



.-. lim. — =/i(.r) = — . 

 AX dx 



If we call 7e^ an infinitesimal that tends towards zero in 

 the same time as a.t, we can write, 



-=/'(.r) + w (9), 



AX 



for taking the limits of both sides, we reach the preceding 

 equation. The variable w is indeterminate and may be 

 plus or minus. Clearing of fractions, we have, 



Ay =/^ (x) AX -\- Zt< AX . . . . (lo). 



In fig. 2, since /^ {x) ax = ax tan IPQ — IQ, we see 

 that the term w Jx must represent IS. 



