The ratio 
AN LSBU 8 Ae B'"B—BB 
BUT DE, PE ANN ae anal Fe toi 2 
IRD abe BB TBB AT 
x 0 10 10 1 0 Ww) 
aN B'B 
En: ete 1 1 2 
Peppy a BeBe 
TO 1 0 
by which the situation of Pir, „Iv is determined, appears to be 
also independent of the charge of the beam. 
But then also the horizontal situations of the other centres of 
2 > 9 De 9 
rotation ern III ,; Pav MWe BV „v are the same for all possible 
charges of the beam. 
For if we consider the two triangles 4,’ B III, and 4,’ B, Il, 
1 
(the latter of which is supposed to bear upon an arbitrary charge 
differing from the one given), in these two affined figures the points 
Pa B and Bae B, Pur, „IV and Pir, Iv are homologous points. 
x x 
From this it can immediately be derived, that also the points 
Ei and Pir, ul, P iv aN and Pav ‚v are corresponding points, 
so that the lines connecting them must pass through the pole of 
affinity, the point at infinity of the straight lines J. 
p) —— . ri =D ¥ * +, 
Pi, u, and Pu, il, as well as P Iv ‚Vv and LAAN wv lie therefore 
perpendicularly above each other. 
From this follows the theorem referred to in the beginning of this §: 
The situation of the centres of rotation Ly Nyt te B, Pir, „IV, 
x 
P 1v vy, relative to the lines /, is quite independent of the charge 
of the beam; it is exclusively connected with the stiffness of the 
beam and that of its supports. 
8. Beam on four points of support. 
When we have once made ourselves familiar with the line of 
thought, developed in the preceding paragraphs, it is rational to try 
and find a solution for the beam on four points of support according 
to the following program. 
1. Cnt the beam at the last point of support but one, and 
construct the situation of the point D in two ways. First by 
determining the reaction yp of the beam CD, freely supported at 
its extremities, and secondly by drawing for the beam ABCD the 
link-polygon belonging to Mc=0O. In this way two points 
