DIFFERENTIAL COEFFICIENT OF AN ARC. 329 



therefore As lies between V{(Ace) 2 + (A^) 2 } and . 



^ 



As ^ 



therefore -T- lies between A /-|l+ f^) ! and 

 Ace 



. / \l + i~ 



V j \A# 



/Ji+/4 



V { \ 



Now the limit of ^/-jl + f/y ) r when Ace is indefinitely 

 diminished, is 



y 



Thelimitof 



/oy\ 2 ) dy dy /f, /oV 



L + hr ' ) r + T- - -7 - , or - 1 - 



V /V/y /I /FT /7 T 



\U/JU/ J \JLJU Ur*JU 



A o /YO 



The limit of - is. by definition, - 7 ; hence 

 Ace c?a? 



as 

 oa; 



Square and multiply by f J , then 



<ds/ \cZs, 

 If x and y are each functions of a third variable t, since 



dx dt , dy dt 



~j~ = "T" > an< * j~ = T5~ > 

 as as as as 



'dt dt 

 we have from (2), (-} = f-? ) + h? J (3). 



