378 Prof. F. A. Tarleton on the Figure of the Bullet which 



to denote the angle made witli the axis of y by the tangent to 

 the curve, and 



dy= | (cos 4 0-3 cos 2 sin 2 0)dO, 



2 



dcc= tan dy— r {cos 3 sin 0— 3 sin 3 cos 0} d0 ', 



hence 



*= ? -f- 1 cos 4 (9- §sin 4 0| + C' = 1 1 -cos 4 0-3sin 4 0} + C 



= - 86 l 4 ~ 4cos26> + 4cos22 ^} + C = ^ cos20 (1 - cos20) + C". 

 Take the origin where = 90, and let - = c ; then C"=2c, 



(CO 



and we have, finally, 



<2?=2c-|-c cos 20(1 — cos 20), 



2/ = c sin 20(1 + cos 20). 

 Prom these equations it can be proved that the curve is a 

 hypocycloid, generated by a point fixed on the circumference of 

 a circle rolling on the inside of another circle with three times 

 as great a radius. 



Take two positions of the moving circle, 0' being the posi- 



