s p i 



If ft be -f-, the spirals begin at the pole, and rreue to r.r. 

 tance j but if n be , the spirals begin at an infinite distance, and r 

 the pole after an infinite number of revolutions. 



2. Spirals to draw tangents to, 

 Subtangent ~ ~~! . 

 Ex. I. In the spiral of Archimedes r-~aS t 



:. Subtangent = , and hence p ~ , 

 V a 4-r* 



Ex. 2. In the reciprocal spiral, 



Snb tangent tt, and p ~ - a 



x. 3, In the logarithmic spiral, 



Subtangent = ~ and p ~ a r, 



3, Spirals to find the areas of, 



Area = fL - . 

 Ex. 1. lu the spiral of Archimedes, 



* A _ *" ** 



SA: 2, In the reciprocal spiral. 



Suppose the area to vanish when r b % then will the area, intercepted 

 ietweea two radii 6 and r, (6 r). 

 Ex. a In the logarithmic spiral, 

 Area between two radii b and r =. ~ (r* *- &*), m being tht modulai. 



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