380 On the Bullet which experiences the least resistance fromthe Air. 



From these equations 20 can be readily eliminated, and the 

 equation of the curve is found to be 



(x* + yy + 2c%{3x*-5y' 2 ) + 12c' 2 a;' 2 -cy + 8c 3 a; = 0. 



If we refer the curve to the three lines joining the cusps, its 

 equation becomes 



u*/3 2 + /3y + ? V - 2«/3 7 2 - 2/3ya 2 - 2ya/3 2 = 0. 



I had worked out these equations and had found the shape of 

 the curve and several of its properties before I knew it was a hy- 

 pocycloid. I mentioned some of its properties to Mr. Burnside, 

 who said he thought it must belong to the cycloidal class. It 

 was easy to see that, if this were so, one circle must have a cir- 

 cumference three times the other ; and on investigation I easily 

 proved, as above, that the hypocycloid generated as described 

 coincides with the curve generating the surface of least resistance. 

 One question still remains, Which branch of the curve is the 

 one of least resistance ? Is it that which meets the axis in a 

 cusp and touches it there, or that which cuts it at right angles ? 

 To solve this question we must recur to theCalculus of Variations. 



Let 



To apply Jacobus test, we have 



7™= - |«n ««? 0(1 -3 tan«0) ; 



VTy) 

 and taking the cusp B for origin from B to A, , v , is posi- 



tive, and hence the resistance is a minimum ; but if we take 0' 



d 2 Y 

 for origin, from 0' to A , g is negative, and the resistance 



( d i) 



is a maximum. Hence one branch of the curve generates the 

 solid of least resistance, and the other the solid of greatest resist- 

 ance, the volume in each case being given. 



The foregoing investigation appears to possess some import- 

 ance in connexion with the inquiry as to the shape best suited 

 for the termination of long-range bullets. Part of it was pub- 

 lished in Major Leech's f Rifle Shooting in Ireland ;' but I was 

 so hurried in the preparation of my chapter in order not to delay 

 the publication that I was unable to complete it as I should have 

 wished. 



