produce Translation and Rotation in the Bores of Rifled Guns. 197 

 M.^f=G + R. {cos v-Atj cosy}, ... (2) 



(3) 



dt*~ Mp* ' 



/o being the radius of gyration. 



We proceed to determine the value of the angles a, @, y, X, /*, v. 

 Let the equations to the helix described by the point P be put 

 under the form 



x=r . cos<£, 2/=rsin<£, z = kr<f>, ... (4) 

 h being the tangent of the angle at which the helix is inclined to 

 the plane of xy. Then 



dx=—r sin <$>d<f>, dy — r cos (j>d(f>, dz = krd(f>, 



and 



ds=rVl + k 2 .dcf> ) 



dx — sin 6 

 cos a = -j- = , 



ds tfl + g 



~ dy cos <b 

 cos£= l 



cos 7 = 



dz _ k 

 ds~~ VT+F 



(5) 



To determine the values of \ p, v, we shall first seek the 

 equation to the driving-surface of the groove. In the case under 

 consideration, the surface is a well-known conoidalone, the "skew 

 helicoid," and is familiar to the eye as the under surface of a 

 spiral staircase. It is generated by a straight line which, pass- 

 ing through the axis of z, always remains perpendicular to it, 

 and meets the helix described by the point P. The equations to 

 the director being given in (4), if x v y^ z 1 be the current coor- 

 dinates of the generator, its equations are 



X\y-y\%=Q, *i=z (6) 



Hence 



z z 



x—r .cos yi, y = r sin -7^ : 

 kr 3 * kr ' 



and the equation to the surface is 



y x . cos ^ — a?, . sin 7^ =0, 



yi kr l kr ' 



or, dropping the suffixes, 



y.cos^-tf.sin-^O (7) 



