98 DIFFERENTIAL AND INTEGRAL CALCULI'S. 



I 



therefore 



dp . . 



-~ x cos ty -f y sin ^ 



+ [-JT sin-vlr ~~ cost/r which = for -f^ = 

 Vtftyr ctyr ofo 



= ZN+MP=YP= perpend, from on the normal at P, 



<P0 



* 2 = x sin y + y cos y 



-I -= cos+ sin-v> which =-- cos 2 -- + sin' 



ds dx , ^ 



= -y since ~^- = cos -^ and ^-=s 

 J^ ^5 efo 



ds 



. 



therefore 



Hence we have s = - T . -f fd\lr, and since , is the 

 ^ rf^r 



perpendicular on the normal at P, we shall have for the 

 whole arc of any closed oval curve without singular points, 

 so that the curve re-enters as soon as ty is increased by 2?r, 



that the whole perimeter = I pd-ty, since will have 



the same value when ty = as when -^ = 2?r. 



Since the normal at P (fig. 23), as P occupies successive 

 positions on the curve, will touch some curve or other, let 

 Q be its point of contact, QY' perpendicular to PQ will 

 be the normal at Q to this curve, and since we have seen 



that -~- is the perpendicular from on PQ, if we call this 

 p ', the perpendicular from on the normal at Q will be 

 -j , but ^'=i7r + ^, and p =. J- therefore the per- 



