NATURAL PHILOSOPHY. 177 



desirable : this may be most distinctly stated as follows : 1. The tube of the 

 spirit-level being a small arc of a circle of considerable radius, within the 

 limits of this arc, the true horizontal line is the tangent to that point of the 

 arc at which the bubble appears ; and in any successive positions, the change 

 of inclination is measured by the angle of intersection of the tangents, which 

 is the same as the arc of the tube traversed by the bubble, or the angle at the 

 centre. 2. Taking the chord of this arc as the base, a plane mirror inclined 

 at 45 to it above, gives for the reflected image of the level-tube a similar arc 

 convex towards it. 3. A lens placed in front of the mirror, having its prin- 

 cipal focal length equal to the distance of the image of the bubble when at 

 the middle point of the arc, and adjusted to receive centrically the reflected 

 diverging pencil from the bubble will cause it to emerge in a parallel pencil, 

 thus placing the image at an infinite distance, and enabling us to view it 

 through a telescope. 4. In other positions of the bubble (since the convexity 

 is towards the lens) it will not be accurately in focus; though it will be ap- 

 proximately so if the focal length be considerable, so that the small differ- 

 ence in the length of the rays from the middle point, and from the two ex- 

 tremities, may be neglected. 5. If the base be accurately horizontal, and 

 consequently the bubble in the middle of the arc to which the index is ad- 

 justed, the axis of the reflected pencil from its image passing through the 

 centre of the lens will be accurately horizontal, or the image seen through 

 the lens gives the true horizontal point; this adjustment is supposed accu- 

 rately made in the first instance once for all. 6. If the base (the lens and 

 mirror being firmly attached to it) be inclined either way within the limits of 

 the arc, any change in the position of the bubble measured on the arc of the 

 tube will be exactly equal to the change measured on the arc of the reflected 

 image. 7. If the focal length of the lens be equal to the radius of the arc of 

 the level, and if the distance were accurately the same from the lens to all 

 points of the reflected arc (as it would be if the image were concave towards 

 the lens), then for any change of arc there would be an exactly equal change 

 of angle in the rays; and as in the middle position the reflected ray is exactly 

 horizontal, so in every other position would the reflected rays for those posi- 

 tions respectively give the tine horizontal point. 8. If the radius or focal length 

 be large and the arc small, the conditions will be so approximately fulfilled, 

 that without sensible error the change of angle will be equal to the change 

 of arc, or of inclination to the horizon, and consequently the reflected ray 

 will give the true horizontal point without sensible error at all inclinations 

 within the limits. Professor C. P. Smyth's invention becomes of peculiar 

 value (in combination \vith his free-revolver stand), since the attainment of 

 an artificial horizon on board ship has been in vain sought in any applications 

 of the ordinary spirit-level, of liquid reflecting surfaces, by simple suspension 

 in equilibrio, or by the rotating plane reflector of Troughton's top, the failures 

 of which are demonstrable on mechanical gi-ounds. But the invention may 

 become of not less importance to observers on land, and especially to scien- 

 tific travellers, from its portability and exemption from the disturbances in- 

 cident to liquid reflectors. 



ON THE COMPOSITION OF ROTARY MOTION. 



At a recent meeting of the Liverpool Philosophical Society, Prof. Hamil- 

 ton read a paper on the " composition of rotary motion," of which the fol- 

 lowing is a resume: 



The principle of compound rotations, he observed, was discovered by Paul 



