This paper is a continuation of my paper, “On a Method of Studying Displacement,” in last year's Proceedings. In that paper I showed how the chief theorems as to the displacements of rigid bodies could be simply demonstrated by the use of what I called a Displacement-chain or Displacement-sequence

The method of extracting any root may be understood from the subjoined example of extracting the fifth root of 2,073,071,593.

The following examples illustrate a somewhat obvious extension of the method of factoring given in a former paper (Proceedings, Vol. XII., p. 32, q.v.). By means of it, if we are given any function of n variables, no one of which is of higher degree than the second, we can either find the factors of it, or prove that it has no factors with rational coefficients.

In the theory of Electrostatics, or of the Newtonian potential, there exists between two systems of potentiating matter, a wellknown reciprocal relation, analytically expressed in the proposition known as Green's Theorrn. By applying his theorem to the case when one of the systems is of the simplest possible character, namely, a mass concentrated at a single point, Green deduced a general method of solving the equation for the potential. The idea of a similar general method of dealing with the equations of Elasticity is due to Professor Betti, of Pisa, who has proved a reciprocal relation between two states of strain of an elastic solid, analogous to the relation in Electrostatics referred to.

I do not think any apology is needed for asking the Society to consider the treatment of Proportion in Elementary Geometry. Although the fifth book of Euclid's Elements appears in all editions of Euclid, I know of no school or college where it is read; I know of no examination for which it is prescribed, and I have never seen an examination paper which contained a question based upon it, except in regard to its definitions. Indeed I believe it is not unfair to say that even among teachers themselves a thorough knowledge of Euclid's fifth book is very rare.