Since (ab) = a l b 2 -a 2 b l, that this may be the case each form must be linear; and if the forms be different (ab) is an invariant (simultaneous) of the two forms, its real expression being **aob** l -a l b 0.

For w = i the form is A i ai+Bib i, which we may write **aob** l -albo = ao(I) b -(I)abo; the remaining perpetuants, enumerated by z I - 2' have been set forth above.

This can be done by placing at B an equal negative point-charge -q in the place which would be occupied by the optical image of A if PO were a mirror, that is, let -q be placed at B, so that the distance BO is equal to the distance AO, whilst **AOB** is at right angles to PO.

But, if the pasteboard be interposed so as to intercept the vibrating segments **AOB**, DOC, the note becomes much more distinct.

The reason of this is, that the segments of the plate AOD, BOC always vibrate in the same direction, but oppo sitely to the segments **AOB**, DOC. Hence, when the pasteboard is in its place, there are two waves of same phase starting from the two former segments, and reaching the ear after equal distances of transmission through the air, are again in the same phase, and produce on the ear a conjunct impression.

32) and the **Aob** l iE0a or AovuaiOa of Ptolemy (v.

Let -y denote the total angular velocity of the rotation of the cone B about the instantaneous axis, $ its angular velocity about the axis OB relatively to the plane **AOB**, and a the angular velocity with which the plane **AOB** turns round the axis OA.

Or because of the proportionality of the sides of triangles to the sines of the opposite angles, sin TOB: sin TOA: sin **AOB**:: a: ~: y, (8 A~

Now, as the line of contact OT is for the instant at rest on the rolling cone as well as on the fixed cone, the linear velocity of the point E fixed to the plane **AOB** relatively to the rolling cone is the same with its velocity relatively to the fixed cone.