8) may be cited as the two most important types, both as regards geographical and geological range, of this Mesozoic family; these ferns are recorded from England, France, Belgium, Germany, Austria, Portugal, Poland and Italy (Map B, M 1), also from Greenland (Map B, **M2**), Spitsbergen (Map B, M3), and Persia (Map B, M 4).

It is an old and easily proved theorem that, for a curve of the order m, the number 6+K of nodes and cusps is at most = Ernr) (**m2**); for a given curve the deficiency of the actual number of nodes and cusps below this maximum number, viz.

Or generally, if M 1 **M2** M3 are the moments of the external forces to the left of 0, A, and B respectively, and s, t and c the perpendiculars from 0, A and B on the directions of the forces cut by the section, then Ss=M11 Tt=**M2andCc**=M3.

With regard to a parallel line through the centroid are given by Mi=N1 - xNo=o, **M2** = N 2 - 2xN i -{- x 2 No = N2 - x2No, M Q = N Q - g _ 1 q(q 2, I) x 2 Ns -2

Then, since Hp, the force at M due to m and - m, is the resultant of d, and - **m2**, we have (L Hp m d d2 2ml H = d3, the direction being parallel to AB.

As the loads move over the girder, the points C, D, E describe the parabolas M1, **M2**, M3 i the middle ordinates of which are 4W 1 1, 4W 2 1, and 4W3l.

P,, we draw any system of parallel planes meeting a straight line OX in the points Mi, **M2**,

Since h = H/m, the inaccuracy in taking C I as the area varies as I /**m2**.

Hence (mi, my, **m2**) is called the actual or effective force on m.

+o (m l **m2** m3) +..., Theorem of Expressibility.