Ex. 4. Inthelitutw, 



Area = c 8 log. . 



4 Spirals to find the lengths of. 

 d z* - d r* 4- r 2 d 6*. 



or dz - T T . (p perpendicular on the tangent) 

 Vr*j* 



Ex. I. In the spiral of Archimedes, 



Are:=z fl. dr ^ a* + -t* t and .'. a parabolic arc, whose 

 latus rectum is 2 a, and whose ordinate is r (see Rectification.) 

 Ex. 2. In the reciprocal spiral, 



Arc arc of a logarithmic curve contained between the ordi- 

 nates b and ;; the subtangent of the curve being equal to the subtau- 

 gent of the spiral. 

 Ex. a In the logarithmic spiral, 

 Arc = VC 1 4- m *) ( *) 

 Ex. 4. In the involute of a circle, 



Arc = (a radius of the circle). 

 5. Spirals t curvature of. 



Had. of curv. = ^ . 



Zpdr 

 Ch. Cttrv . = _-. 



Ex. 1. In the logarithmic spiral, 



Rad. curv. = , and ch. curv. =2r. 

 in 



Ex. 2. In the spiral of Archimedes, 

 Ha,eu r ,=^^ 



Ex. 3. In the recif rocal spiral, 



2r (a +r*} 



Ch. curv. J ~ - ; . 



a 



S72 



