ON THE AREA OF THE CYCLOID. 225 



directrix and of rotation round its own centre. The area gene- 

 rated by the describing point may be considered as generated 

 by these two motions : that of translation nowise affects the 

 motion of rotation, and the area due to the latter is the same 

 as if the former did not exist, that is, it is equal to the area 

 of the generating circle/ Contrariwise the motion of rotation 

 does affect the area due to that of translation, inasmuch as in 

 virtue of it the distance of the describing point from the di- 

 rectrix is varied : the mean distance, viewed as depending on 

 the motion of rotation, is equal to the radius of the generating 

 circle, and the corresponding area is therefore a rectangle, the 

 base of which is the space slided over and altitude that radius ; 

 and, as this space is the circumference of the generating circle, 

 the area in question is equal to twice the area of that circle : 

 on the whole, therefore, the area of the cycloid is equal to three 

 times that of the generating circle. 



" The reason is just the same as that by which what are 

 called Guldinus's properties are established. We here resolve 

 the motion of a describing point into motions parallel and 

 perpendicular to the abscissa; the latter generates no area, the 

 former generates a rectangular area having for its base the 

 abscissa and for its altitude the mean value of the ordinates ; 

 that is, the ordinate of the centre of gravity of the arc, which 

 is a known result. The only difference to be attended to in 

 the two cases relates to the mode in which the average is to 

 be taken." 



Mr Ellis has remarked, that the same method may be ex- 

 tended to the determination of the areas of the hypocycloid and 

 epicycloid. 



I am, Sir, 



Your obedient Servant, 



WILLIAM WALTON. 



Cambridge, July 31, 1854. 



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