178 EXAMPLES OF THE CHANGE 



Put n = 1 ; then 



dt / dx dx*' 

 dy dy dx dy 



TD..A H_ 7 /j. _Z 



<& ofa dt dx } 



a cPy (d \dy ,. 



therefore a; -r , = (-r. - 1 ) -77 (2). 



- -j - - 

 dx \dt J dt 



Put n = 2 in (1) ; then 



dt J dx* 

 or from (2), 



/ 



Proceeding thus we deduce 



201. It is often useful in geometrical applications of the 

 Differential Calculus to have expressions for -~ and -~ in 



terms of 6, supposing 



x = r cos 6 \ 



y = r sin 6 } " 



Since y is by supposition some function of x, it follows 

 from (1) that an equation subsists between r and 6, so that 

 r may be considered some function of 6. 



d y ; /i*i 



dy dff M 



Now -& - TX =:& from 



-JH cos 6 -jr. r sin 

 dQ dv 



ffiy 



_ 



dx* dQ~ n dr ~ dx 



cos - r sm 



