122 CELESTIAL MECHANICS. I. ii. 9. 



coinciding in length and in curvature with that of the circle 

 of equal curvature at the given point. ' . 



^69' Lemma C. The radius of curvature 

 is in general r=da: : d ^ = — d?/ : d —. 



Since d sin 9zz cos dd9 (142), we have d6= ; and 



cos 9 



Since d cos d zz — sin 9a9, d9 zn . Now sm d= 



sin 9 



dy J - djT , 1/, J dy d* , dx ds 



^, and cos5i=~; whence d9=i d-^. — = — d — . — , 

 u* d* ds dx ds dy 



J d^ J J dy - . dx 



and r=:-r- i=dj7 : d -^ = — dy : d -7-. 

 d9 ds ^ ds 



270. Lemma D. When the fluxion of the 

 curve is constant, the radius of curvature is 



r- 



__d«da7__ d5dy__ ds^ 



'""dd^"" dd^~V(d^^^M^^V)* 



When ds is constant, we have d --=^=--^, and d -7- = 



d* d* d* 



dda: , , , d?/ dsda: dsdy t» ^ . , „ 



--— , and rzzdz : d -^z=--_- = — ^ . iJut since dx^ 



ds ds ddy ddj; 



-^dy^zzds^, dxd^x + dyd^yzzO, and dx^d^x^zzdy^d^y^, 



consequently j^^ = ^, and ^.^e+^y = dI?Td^ 



,-.^ dor^ , ddv dor , d^dor 



(34) = — — , and ; —- r==-T- v whence — r-r- = 



^ ^ d52' ^/(d2a72+d2y2) d5 ddy 



d«2 



V(d2a;2+d2y2y ^ 



271. Lemma E. When the curve is re- 

 ferred to three orthogonal ordinates, its fluxion 



