398 Mr DAVIES on the Equations of Loci 



The values of the differential coefficients derived from (8) being inserted in 

 (15), give the cosine of the radius of curvature of (8) at the point <p 6. In- 

 serted in (14), they give the polar distance of the centre of curvature : and 

 inserted in (16) next following^ they give the longitude of the centre of cur- 

 vature. 



Resume (12), expand it, and arrange according to sin K and cos K : then 

 we have 



cos K ( A cos 6 + B sin 6) + sin K ( A sin 6 B cos 6) = 0, 



A cos 6 + B sin 6 tiu\ 



ortan/c = , ^ (lo.) 



B cos u + A sin 6 



The magnitude and position of the circle of curvature is hence determined 

 when we insert from the given equation (8), the values of the differential 

 coefficients in the forms above found. 



It will be remarked, that these results suppose both variables to be depen- 

 dent upon some third variable ; that is, they are such as that the first dif- 

 ferential of either does not become constant. We may, in general, then, 

 very greatly simplify these expressions by taking either d* $ or d* 6 equal 

 to zero, inasmuch as B will then become either 



cos (p d 6 d (p + sin (p d* 6 cos(pd6d(p s d? (j) d 6 sin (p. 



sin <pd<p* sin(f>d (p* 



It is unnecessary to pursue this subject farther here, or we might effect a 

 considerable simplification by the reduction of the denominator of (15) to 

 another form. We shall, therefore, finally, make one remark concerning 

 the evolute or locus of the centre of curvature. 



This may be considered either as the locus of the centre of curvature, or 

 as the locus of the intersections of the consecutive normals. 



Let /(00) = (17.) 



be the curve whose evolute is sought. 



Then if between (14, 16, 17), we eliminate and 6, we shall have an 

 equation involving only K and A with constants, which will be that of the 

 evolute required. 



If we consider it as the locus of the intersections of the consecutive nor- 

 mals, we must proceed thus : 



