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„force acting at this extremity, the execution of the required con- 
struction would only increase the difficulty mentioned at the begin- 
ning of this §. 
Yet all the auxiliaries for a fit construction of the elastic link- 
polygon of a beam on five or more points of support have been 
provided, as will appear from the following. 
13. Beam on more than four points of support. 
Let the beam on n points of support A, B. C,... U, V, W,- be 
given, and let it for the present be required to determine the descent 
and the inclination at the last point of support W, when the beam 
is successively charged at W by a unit force and a unit moment. 
Then the experience gained in the preceding $$ leads to the expec- 
tation that the quantities in question only depend upon the corre- 
sponding ones for the beam A, B, C,... U, V, i.e. upon the descent 
and the inclination which will appear at the last point of support 
of the beam A, 5,C,...U,V, when in V a unit force, resp. a 
unit moment, acts. 
Let us suppose these latter quantities, which may be indicated by 
To Pn—2, hs Gn—2, for a moment to be known, and let us attempt 
to derive from them the ys, Pa—1, OE Pea required. In deter- 
mining each of these quantities we might again use the introduction. 
of different moments of transition My above V. 
For each moment of transition My it would be necessary to 
determine the situation of the point W in two ways: 
1. by the aid of the equations of equilibrium of the field V W 
to the right, supposed to be free, by means of which a point W 
is found; . 
2. by means of an elastic link-polygon belonging to the beam 
A, B, C,... U, V, W, which gives a point W. 
If then the series of points W and W should appear to be similar, 
it would be possible to construct their double point, i.e. the extreme 
point of the link-polygon determining the required quantities. 
If the line of action described really causes y,—1, Pnr—1, Veer 
ie to be found, we must accordingly attempt to determine the 
corresponding quantities of the beam ABC...UV on (n—1) points 
of support. 
But this would be possible if under the same conditions of charge 
the inclination and descent were known for the beam ABC... UD 
on (2-2) points of support. 
On arguing further in this way, we are driven back to the beam 
