CONTACT OF POLAR CURVES. 343 



The chord of curvature passing through the origin will be 

 obtained by multiplying 2p by the cosine of the angle be- 

 tween the radius vector and the normal to the curve at the 

 point considered. (Art. 320.) Hence the chord of curvature 

 through the origin 



324. If ty be the angle which the tangent at the point 

 (x, y] of a curve makes with the axis of x, we have 



d 2 y d*y 



therefore d = ^ 2 ^ = ^ 



\dx/ I \dxj 



therefore o = 1 . 



325. If two polar curves have a common point the polar 

 co-ordinates of that point must satisfy the equations to both 

 curves. If they have contact of the first order at that point 



the value of -~ is the same for both curves at that point, and 



dx dr 



hence, by Art. 201, the value of -^ is the same for both 

 J dd 



curves. If the curves have contact of the second order the 



d*ii 

 value of -j~ also is the same for both curves at the common 



dx d*r . 



point, and hence, by Art. 201, the value of -^ is the same 



for both curves at that point. Proceeding in this way, we 

 see that if two curves have contact of the n lh order at any 

 point, if they are referred to polar co-ordinates, the values of 



