406 APPLIED MECHANICS 
the cross are jointed to the forks, so that they may turn freely about their 
axes. As the shafts rotate the axis of aa, describes a circle whose plane 
is perpendicular to the axis of 
the shaft c, and the axis of 00, 
describes a circle whose plane 
is perpendicular to the axis of 
the shaft d, and as the axes of 
the shafts are assumed to be 
horizontal, the planes of these 
circles are vertical, Referring 
now to the upper part of Fig. 
657, the circles described by 
the axes of the cross are shown 
projected on a plane perpendi- 
cular to the axis of the shaft c. 
The circle described by the axis 
of aa, projects into an equal 
circle AXA,Y, and the circle 
described by the axis of bb, 
projects into an ellipse YB,B, 
the semi-minor axis of which 
is equal to rcos?, where r 
is the radius of the circles 
described by the axes of the 
cross. 
If OA be the projection of the axis of the arm oa carried by the shaft 
c, then OB, the projection of the axis of the arm ob carried by the shaft d, 
must be perpendicular to OA, since these axes are perpendicular to one 
another, and they are projected on a plane containing one of them. Also, 
when OA has turned from the horizontal position OX through an angle — 
a, OB will have turned through an equal angle « from the vertical position 
OY. But the actual angle 8 through which the arm od has turned is not 
the angle BOY but the angle B’OY, the point B’ being on the circle 
AXA,Y and in a.line B’BN perpendicular to OY. 
The connection between a and f has now to be found. 
/ 
BN 
BN =B’N cos 9, hence ON = ON © 6, therefore tan a= tan B cos 0. 
Fic. 657. 
The angular velocity w, of ob at any instant is evidently not neces- 
sarily the same as w,, the angular velocity of oa at the same instant, and 
the ratio of these two angular velocities will be the ratio of the indefinitely 
small increase of 8 to the corresponding increase in a. Differentiating 
h : _ dp se®a 
the equation tan a= tan f cos 6, the result is da wae po, 
cos 0 
— sin? 6 cos? a. * 
Eliminating £, this reduces to ir 
My « : 1 ; 
er has a maximum value = ae when cosa=1 or—1, that is, when 
a=0° or 180°. 
°> has a minimum value=cos@ when cosa=0, that is, when 
Oy 
a= 90° or 270°. 
