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 L2 10s. or L3 may be said to satisfy this first condition.  The price for the first complex integraph designed by Coradi was L24 to L30.  The modified form in which I show it to-night is estimated to cost retail L14.  Till an equally efficient instrument can be produced for L5 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.

2.  The instrument must not be liable to get out of order by fair handling and a reasonable amount of wear and tear.  I cannot speak at present with certainty as to how far our integraph satisfies this condition; it is rather too complex to quite win my confidence in this respect.

3.  It must be capable of being used on the ordinary drawing board, and of having a fairly wide range on it, i.e., it must not be limited to working where the primitive is at one part only of the board.

This condition takes out of every day practical drawing use the integraph invented by Professors James and Sir William Thomson, in which the sum curve is drawn on a revolving cylinder.  It is essential that the sum curve should be drawn on the board not far from the primitive, and that this sum curve can be summed once or twice again without difficulty.  The time involved in drawing the four sum curves,