S3 2 DIFFERENTIAL COEFFICIENT OF AN ARC. 



ds /(fdr^ 

 therefore -^ 



ds ds dd 



Also -r- = -jx -5- = A / -u-t-r i j- j f. 



rfr eft? c?r VI \" r / J 



We have shewn in Art. 279, that 



d0 



tan <> = r -y- , 

 dr 



where <j> is the angle between the radius vector at the point 

 whose polar co-ordinates are r, 0, and the tangent at that 

 point. Hence 



dO dO 



dr dr dO 



cu 

 dr 



ds ds ' 



Oar 



Similarly cos <f> = -r- . 



These results may also be deduced immediately from the 



PL 

 figure in Art. 279; for sin (j) is the limiting value of -p-~., 



,1 , c PL As ,, rsinA0 As , T . 



that is, of ^-.737-, or of Am .7^. The limit 



. , ,, v ., ,, As . . 



is } ; and the limit of -7 is unity ; hence 



As ds 



sin <j> = -j- . Similarly the value of cos < may be found. 



ds 

 311. The value of -^, in Art. 310, may also be obtained 



thus: 



Let P, Q, be points on a curve, and suppose 

 sp r PRrQ 



O-i ~ / * JT OfcO l/j 



Draw PL perpendicular to SQ, 

 then 



PL = r sin A0, 



