By Professor KARL PEARSON, M.A.

As I fear the title of my paper to our Society to-night contains two misstatements of fact in its three words, I must commence by correcting it. In the first place, the instrument to which I propose to draw your attention to-night is, in the narrow sense of the words, neither an integrator nor new. The name "integrator" has been especially applied to a class of instruments which measure off on a scale attached to them the magnitude of an area, arc, or other quantity. Such instruments do not, as a rule, represent their results graphically, and we may take, as characteristic examples of them, Amsler's planimeter and some of the sphere integrating machines.

An integrator which draws an absolute picture of the sum or integral is better termed an "integraph." The distinction is an important and valuable one, for while the integraph theoretically can do all the work of the integrator, the latter gives us in niggardly fashion one narrow answer, et praeterea nil. The superiority of the integraph over the integrator cannot be better pointed out than by a concrete example. The integrator could determine by one process, the bending moment, from the shear curve, at any one chosen point of a beam; the integraph would, by an equally simple single process, gives us the bending moment at all points of the beam.

In the language of the mathematician, the integrator gives only that miserly result, a definite integral, but the integraph yields an indefinite integral, a picture of the result at all times or all points - a much greater boon in most mechanical and physical investigations. Members of our Society as students of University College have probably become acquainted with a process termed "drawing the sum curve from the primitive curve." Many have probably found this process somewhat wearisome; but this is not an unmixed evil, as the irksomeness of any manual process has more than once led to the invention of a valuable machine by the would-be idler. Thus our innate desire to take things easy is a real incentive to progress. It was some such desire as this on my part which led me, three years ago, to inquire whether a practical instrument had not been, or could not be, constructed to draw sum curves. Such an instrument is an integraph, and the one I have to describe to you to-night is the outcome of that inquiry.

It is something better than my title, for it is an integraph, and not an integrator.

A NEW INTEGRATOR.

Before I turn to its claims to be considered new, I must first remind you of the importance of an instrument of this kind to the draughtsman. I put aside its purely mechanical applications, where it has been, or can be, attached to the indicators of steam engines, to dynamometers, dynamos, and a variety of other instruments where mechanical integration is of value. These lie entirely outside my field, and I propose only to refer to a few of the possible services of the integrator when used by hand, and not attached to a machine.

The simple finding of areas we may omit, as the planimeter will do that equally well. But of purely graphical processes which the integraph will undertake for us, I may mention the discovery of centroids, of moments of inertia (or second moments), of a scale of logarithms, of the real roots of cubic equations, and of equations of higher order (with, however, increasing labor). Further, the calculation of the cost of cutting and embanking for railways by the method of Bruckner & Culmann, the solution of a very considerable number of rather complex differential equations, various problems in the storage of water, and a great variety of statistical questions may all be completely dealt with, or very much simplified by aid of the integraph.

In graphical statics proper the integraph draws successively the curves of shear, bending moment slope, and deflection for simple beams; it does the like service for continuous beams, after certain analytical or graphical calculations have first been made; it can further lighten greatly the graphical work in the treatment of masonry arches and of metal ribs. In graphical hydrostatics it finds centers of pressure and gives a complete solution for the shear and bending moment, curves in ships, besides curves for their stability. In graphical dynamics the applications of the integraph seem still more numerous. It enables us to pass from curves of acceleration to curves of speed, and from curves of speed to curves of position. Applied to the curve of energy of either a particle or the index point of a rigid body, it enables us by the aid of easy auxiliary processes to ascertain speeds and curves of action. In a slightly altered form, that of "inverse summation," we can pass from curves of action to curves of position, and deal with a great range of resisted motions, the analysis of which still puzzles the pure mathematician; the variations of motion in flywheels, connecting rods, and innumerable other parts of mechanism, may all be calculated with much greater ease by the aid of an integraph.

Shortly, it is the fundamental instrument of graphic dynamics.

It would be needless to further multiply the instances of its application; the questions we have rather to ask are: Can a practical instrument be made which will serve all these purposes? Has such an instrument been already put upon the market? If I have to answer these questions in the negative, it is rather a doubtful negative, for the instrument I have to show you to-night goes so far, and suggests so many modifications and possibilities, which would take it so much further, that it is very close to bringing the practical solution to the problem.

Let me here lay down the conditions which seem essential to a practical integraph. These are, I think, the following:

1. The price must be such that it is within the reach of the ordinary draughtsman's pocket. The Amsler's planimeter at £2 10s. or £3 may be said to satisfy this first condition. The price for the first complex integraph designed by Coradi was £24 to £30. The modified form in which I show it to-night is estimated to cost retail £14. Till an equally efficient instrument can be produced for £5 I shall not consider the price practical. If the error of its reading be not sensibly greater than that of a planimeter, it is certainly worth double the money.