28 PROFESSOR FLEEMING JENKIN’S APPLICATION OF GRAPHIC METHODS 
come at od. The diagram is drawn without taking friction or stiffness into 
account, so that the frame shown is kinematically equivalent to the machine. 
It would be easy in a drawing on a large scale, to show the complete effect of 
friction and stiffness, for we have already learnt to take these into account for 
each component simple machine. We see, therefore, that in any simple train 
of machinery, there can be little difficulty in estimating the true relation 
between effort and resistance (neglecting weight and mass) ; this difficulty never 
exceeds that met with in analysing a simple machine, and all simple machines 
are of one type. 
§ 25. Half Machines Compounded.—In certain cases two successive machines 
have two elements in common, as well as the driving or resisting link. In this 
case we may consider the addition as in reality only half a machine. The 
typical example of this arrangement is given in fig. 30, where links 6 and ¢ are 
common to two complete machines—Ist, abcde, with its resisting link; and 2d, 
behgi, with its driving link. It will be seen that the half machine hig has 
a certain analogy with the machines of class 2, only it is here the stiff bar 
which acts as a driving element by an alteration in its length; h, 7, or g might 
represent the final resisting element. We are never driven to adopt this sub- 
division of a machine, except at one or other end of a train of machines which 
may not be divisible into a series of complete machines with joints or elements 
of transmission. Thus, if we -(fig. 31) have a steam-engine with a spur-wheel 
on its crank shaft, driving a single pinion from the shaft of which a weight is 
hanging, we cannot divide the train into two distinct complete machines, 
each having a quadrilateral with diagonals as its dynamic frame. We may, 
however, draw two distinct dynamic frames, as shown in fig. 3la, where 7, 8, 9, 
10, 11, and 12, are the links of a complete spur-wheel machine, similar to that of 
fig. 22a, driven by a link 7, but link 7 is in the same line as link 2 of the well- 
known engine frame. Links 8 and 38, 6 and 11, are also common to the two 
frames. The diagram shows that we might analyse the machine in two ways, 
calling, for instance, the steam engine a complete machine, and the extra spur- 
wheel with its weight, a half machine ; or we might call the two spur-wheels a 
complete machine, and the piston, steam, and connecting-rod a half machine. 
It is a matter of no consequence how we subdivide a train. The relative 
stresses in the first and last links will be the same, whether we use half 
machines with two common elements, or successive machines with one common 
element. This example shows us that the machines of class 2 may very 
properly be regarded as only half machines. This view is supported by the 
observation that when one machine of class 2 drives another of the same class, 
we get for the systema single complete dynamic frame, namely, the quad- 
rilateral with two diagonals. As an example, we may take the hanging pulley 
and fixed pulley combined, as in fig. 32, on which is shown the dynamic frame of 
