THE ROYAL ARTILLERY INSTITUTION. 
411 
and 
dx — sin d> , 
cos a “ — = I 
ds Vl + ti* 
COS /3 = -|- = 
cos y = —• = 
COS 1$ 
vT+1 
Jc 
ds VT+~¥' 
(5) 
To determine the values of A, ju,, 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 conoidal one, 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, passing 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 y v z x be the. current co-ordinates of the generator, 
its equations are 
— Vi x = 0* z ± — z .(6) 
Hence 
. z, 
x — r cos , y = r sm -l ? 
kr hr 
Now A, , v being the angles which the normal to (7) makes with the axes, 
[VOL. III.] 
\dxj_ 
{dF\* 
{©'+1 
<dF\ 
\dy) 
fdF 
' dy j 
'* (?)? 
) 
{©■♦< 
f dF\ 
>dy) 
/dF 
\ dzj 
’+ (ST 
) 
__ \±l _ 
K dF\* , /dF\* /dF\*)t’ 
s)+U) + (s)} 
(8) 
PP 
