MISCELLANEOUS PBOPOSITIONS. 415 



Tt 



Now -^ is infinitesimal when r is ; therefore, denoting by 



77 an infinitesimal angle, we see that (3) has four different 

 solutions for 6, namely, one between a 77 and a + 77, one 

 between /3 ij and j3 + i), one between TT + CC 77 and Tr+a+i;, 

 and one between TT + /3 77 and TT + /3 + 77. Thus the singular 

 point is a double point, the tangents at the point being in- 

 clined at angles a and /3 respectively to the axis of x. 



386. Next suppose that D 2 is less, than CE ; then we 

 shall find that the infinitesimal circle does not cut the curve, 

 and so the singular point is a conjugate point. 



387. Finally, suppose that J7=CE; then equation (2) 

 takes the form 



o7? 



jE cos 2 (tan 0- tan )*+ = () (4): 



the discussion of this form is rather complex, and we will 

 only briefly indicate the results. 



R 



Suppose that -^ is negative when is indefinitely near 



to a. Then denoting by 17 an infinitesimal angle we see that 

 (4) has two solutions for 6, namely, one between a 77 and a, 



and one between a + 77 and o. The sign of -p when 6 is 



indefinitely near to TT + a will in general be contrary to the 

 sign when Q is indefinitely near to a, because R is in general 

 a function of the third degree in cos and sin 0, when r is 

 small enough ; and so there is no solution of (4) in this case 

 besides the two already noticed. Hence the infinitesimal 

 circle cuts the curve at two points, and only at two ; and the 

 radii of the circle drawn to the two points include an in- 

 finitesimal angle. Therefore the singular point is a cusp; the 

 tangent at the cusp is inclined to the axis of x at an angle a, 

 and the two branches are on opposite sides of the tangent. 



ry 



Similarly if -^ is positive when 6 is indefinitely near to a 

 Mi , 



we have in general a cusp of the first kind as before; the 

 tangent at the cusp is now inclined to the axis of x at an 

 angle TT -f a. 



