experiences the least resistance from the Air. 379 



tions of the generating point ; then the arc 



TO=arcTO'; .'. L 0'C'T = 3Z A T. 



Take A for axis of x, and the tangent to the two circles at for 

 axis of y, A is clearly the tangent to the hypocycloid at the cusp 

 0. The tangent at 0' is perpendicular to ; T, and therefore pa- 

 rallel to C M drawn to bisect 0' T from the centre C. Let the 

 angle OAT = f,ZO' C'T = 3f Again, j^»=Z M C'T = f + 

 angle between MC and 0\A ; therefore this latter L =^ . Hence 

 the angle between the tangent at any point and the tangent at 

 the cusp =o^« The coordinates of 0' are a^OX and y = OY, 



tf = 0A--0'R-PA=3r- 

 2/ = C'P-C'R=2rsinf- 



rcos 2-vJr— 2rcos yjr, 



rsin2-»/r; 



so may be written 



4r— r(l + cos 2yjr) — 2r cos ^r=4r— 2r cos^r(l + cos^r), 



y == 2r sin yfr(l — cos yfr) . 



Now let be the angle between the tangent at 0' and the axis 

 of y } then 



= 90°-^; .'. ^=18O°-20, 

 hence 



x = 4r + 2r cos 20(1 - cos 2(9), 



y = 2r sin 20(1 + cos 20). 



But these are the same equations as found for the generating 

 curve of the solid of least resistance if 2r be assumed equal to c. 



The shape of the curve 

 is therefore evident. It 

 consists of three symme- 

 trical branches, and has 

 three cusps ; moreover 

 the tangent at a cusp is 

 the axis of revolution of 

 the required surface, and 

 the lines joining the cusps 

 form an equilateral tri- 

 angle. 



In the subjoined figure 

 the origin may be sup- 

 posed to have been taken 

 at the point B. If the 



origin be moved to the 



poiut 0', we have 



x = c cos 20(1 -cos 20), 

 z=c sin 20(1 + cos 20). 



2C2 



