308 
MR. R. S. HEATH OH THE DYNAMICS OF 
\(«f —■/&,)=— f{) 
\{bg'-gb')=ix(bb'-gg) 
Hence eliminating X : y 
(b'f — a g') {ab -fy) = (ab' - fg') {bf- ay ). 
This equation must be expressed in point coordinates. Since c = 0 and h=0, 
x' y' 
—=z——m say, 
x y 
and 
z' v! 
—=—=n say. 
z u J 
Substituting for x, y', z , v! in the expressions for a, b,f, y, the factor ( m—n ) 2 divides 
out and the equation to the Cylindroid becomes finally 
(b'f— ag')ccy(z 2 +w 2 ) = (ab / —fg')zw(£c 2 +?/ 2 ). 
This is a surface of the fourth order having the common perpendiculars of the axes 
of the two given screws, for nodal lines. 
38. To find the conditions that any number of twists about given screws may 
produce rest. 
The method here employed is the same as that given by Spottiswoode, in the 
‘ Comptes Kendus,’ t. lxvi. 
Let the coordinates of the screws be a 0 , b 0 . . . a 1? bj . . . , there being n screws; and 
let the magnitudes of the twists on them be £l 0 , Then the conditions 
that they will neutralise each other will be 
2(Ha)=0, X(nb)=0 . . . 2(flh) = 0. 
The expression 
a </i+Kg’i+c 0 h! + f 0 a x +g 0 b 1 + l^cq 
is called the simultaneous invariant of the two screws 0 and 1, and it will be denoted 
by (01). Similar expressions will apply to the other screws. The quantity (00) will 
be the parameter of the screw (0). 
By means of the equations of condition we get 
X(na)X(nf )+1 (nb)2(flg)+s(nc)S(nh)=o. 
Multiply these out, then 
+pih 2 (v)=0. 
Here X implies summation from 0 to n — l inclusive. 
