40 R. G. WEST ON LISSAJOUs' CURVES. 



after the other, by making their times of vibration unequal, or by 

 combining the two expedients, the result is more complicated. If two 

 equal pendulums be started, the one half a vibration after the other, 

 a circular spiral, and with any other fractional start an elliptical spiral, 

 more or less oblong, will be produced ; thus the ratio of 1:2 (illus- 

 trative of the musical interval of the octave) gives, with a simul- 

 taneous start, an approximation to a parabolic curve ; with the half- 

 vibration start, the figure 8. With the higher ratios (2:3, 3:4, 4:5, 

 &c), the effect of throwing the vibrations " out of tune," as it is 

 called, so as to let one pendulum gain on the other, is to pro- 

 duce a more or less confused figure ; but, with the simple ratios of 

 unison and 1:2, very curious and beautiful effects are thus ob- 

 tained, resembling in some cases the markings of Diatoms. By a 

 very slight discordance in the vibrations, however, u water-markings" 

 are produced, which constitute the chief beauty of these curves. 

 Similar "water-markings'' are obtained by the superposition of 

 figures as nearly similar as can be traced, and in both cases the ex- 

 planation lies near the surface. With the unison made so discordant 

 as 29:30, a very curious stereoscopic effect is produced, resembling 

 a piece of network hanging in folds. 



But with the apparatus which I have described, none of the 

 figures are strictly symmetrical, all being resultants of two arcs and 

 not of two straight lines. The distortion, however, may be largely 

 reduced by lengthening proportionately the connecting arms. It 

 was entirely obviated in the Halle apparatus referred to above, 

 an instrument designed by Herr P. Schasnemann, and made 

 by Kleeman, of Halle, in which the tracing-point was actuated by one 

 pendulum only, while a glass tablet, carrying the card, and suspended 

 by an elaborate system of cords, was impelled by the other ; by 

 varying the angular inclination of the vibrations, curious skew and 

 scroll-like figures were produced. Beautiful spiral figures are also 

 obtained by rotating the card by clock-work. 



With the subject of microscopic ruling and writing, I cannot deal 

 completely, owing to the scattered nature of its literature. The finest 

 rulings, prior to 1851, which Ihave found recorded, are those of Fraun- 

 hofer's fixed net micrometer for the telescope, and Sir John Barton's 

 rulings on steel, which were both at the rate of 10,000 to the inch. 

 Impressions of the latter were taken in gelatine, to serve as micro- 

 meters. In 1851, Mr. De la Rue exhibited 10 groups of rulings varying 

 from 11,261 to 56,306 to the inch. Mr. W. Chandler Roberts, of 



