424 THE THEORY OF SCREWS. [388 



Let p be the pitch of the screw on which the wrench thus represented 

 lies, and let a, y, z be the co-ordinates of any point on this screw. Then, 

 in the plane of Z the moments of the forces are xY yX, and if to this be 

 added pX, the whole must equal H. 



Thus we have the three equations, so well known in statics, 



F=pX+yZ -zY, 

 G=pY+zX-xZ, 

 H=pZ + xY -yX. 



The centrifugal acceleration on a point P is, of course, co 2 PH, where o&amp;gt; is 

 the angular velocity, and PH the perpendicular let fall on the axis. The 

 three components of this force are X y Y , Z , where 



X = ft&amp;gt; 2 sin 6 (x sin y cos 0}, 

 Y = ft) 2 cos 6 (y cos 6 x sin 0), 

 Z = ft) 2 0-rasin 2(9), 



and the three moments are F , G , H , where 



F = to 2 sin 9 (yz sin + xz cos 6 Zmy cos 6), 

 G = ft) 2 cos 6 ( yz sin 6 xz cos 6 + Zmx sin 0), 

 H = to 2 1(2/ 2 - 2 ) sin cos 6 + xy cos 20}. 



We are now to integrate these expressions over the entire mass, and we 

 employ the following abbreviations ( 324) : 



jxdm = Mx jydm = My ; $zdm = Mz ; 



jxydm = M1 3 2 ; fxzdm = Ml.? ; jyzdm = Ml? ; 

 X = fX dm , Y=fY dm; Z = jZ dm; 

 F = fF dm; G=fG dm; H=JH dm; 

 then, omitting the factor Ma?, we have 



X= + (a? sin 6 y cos 6) sin 0, 

 Y= (x sin 6 7/ cos 6) cos 0, 

 Z = Z Q m sin 20 ; 



F = + sin (I, 2 sin + I* cos 0) - 2my sin cos 0, 

 G = - cos (^ sin (9 + , 2 cos 0) + 2mx sin cos 0, 

 #= (/&amp;gt;i 2 - p*} sin cos + r- cos 20. 

 We can easily verify that 



