POLAR FORMULA. 305 



Let Q be another point, the co-ordinates of which are 

 r + Ar, and + A#. 



Draw PL perpendicular to SQ, then 

 PL = r sin A0, 



= r + Ar r cos A0 ; 

 rsinA0 



r rnr> 



therefore tanZQF= 



. 



r + Ar r cos 



Let $ move along the curve to P; the limiting position 

 of QP is by definition the tangent to the curve at P; let this 

 be PT. The limit of the angle L QP will be the angle SPT; 

 call this angle <, then 



, ,, ,. ., f rsinA0 

 tan <p = the limit oi - r - T-TT 

 r + Ar r cos Ac/ 



when A# and Ar are indefinitely diminished. 



r sin A# 



r sin A# A0 



INow 



r + Ar r cos A0 , A$ 



2rsm 



2 Ar 



The limit of ,. is 1. 



A.y^ Cf /* 



The limit of - is denoted by - . 



a 

 2 sin sin ^ 



The limit of A ,. , that is, of . a sin - - , is zero. 



At/ iA(7 2 



Therefore tan <f> = r -=- . 



rfr 



T. D. C. 



