416 MISCELLANEOUS PROPOSITIONS. 



But it may happen that R itself changes sign when 6 is 

 indefinitely near to a or to TT + a ; and then our conclusion 

 as to a cusp of the first kind does not hold. We should have 

 in such a case to make a closer examination, and in general it 

 would be necessary to extend our expansion of<j>(x+h, y+ty, 

 and instead of R to have terms which may be expressed as 



where t represents a proper fraction. 



388. Moreover if C, D, and E all vanish at the point 

 (x, y}, we should have to use this extended form of the ex- 

 pansion of < (x + h, y + k} in order to determine the nature 

 of the singularity. 



MISCELLANEOUS EXAMPLES. 



1. If a semicircle roll along a straight line, the curve to 



which its diameter is always a tangent is a cycloid. 



2. If a cycloid roll along a straight line, the equation to 



the curve which its base touches is 



3. A series of circles is described having their centres on an 



equilateral hyperbola and passing through its centre, 

 shew that the locus of their ultimate intersections will 

 be a lemniscate. 



4. Examine the nature of the following curves at the origin : 



y* + 2ay"x + x* 2ax* - 0, 



y* key (ay bx) a; 4 = 0, 

 = 2a*x*y. 



