NA TURE 



[_ June 26, 1884 



instruments, and although the wheels and 

 pinions were made by a>good watchmaker, 

 still the friction involved in such a plan 

 has induced them to abandon it in favour 

 of the new arrangement which is the sub- 

 ject of their present communication. 



The telescopic method employed by 

 Weber, and the spot of light method due 

 to Sir W. Thomson, for magnifying the 

 effect of an angular motion are, of course, 

 unequalled for stationary measuring-instru- 

 ments, but for instruments which must be 

 carried about and used quickly without 

 the necessity of adjustment, these most 

 ingenious reflecting methods are quite- 

 unsuitable. 



With an ordinary cylindric spring, 

 having a small angle between the oscu- 

 lating plane and a plane perpendicular to 

 the axis, as is the case with all spiral 

 springs such as are in practical use, it is 

 well known that but very little rotation is 

 produced between its ends by the appli- 

 cation of an axial force. Consequently 

 with such springs it is only possible to 

 obtain magnification by the employment 

 of a system of levers, or of a rack and 

 pinion. It occurred to the authors, there- 

 fore, to consider whether it would not be 

 possible to make a spiral spring of such a 

 nature that for a comparatively small axial 

 motion of its ends there should be con- 

 siderable rotation of one end relatively to 

 the other, and by the employment of which 

 all levers, racks, and pinions could be dis- 

 pensed with, so that no error could be 

 introduced by wear and tear, or by want 

 of fitting of joints, and further so that the 

 temperature correction should be merely 

 one affecting the frigidity of the material 

 used as a spring, and not a correction such 

 as had to be applied in consequence of 

 the contractions and expansions of the 

 various parts of an ordinary magnifying 

 apparatus. 



The theory of the strength and stiff- 

 ness of the ordinary cylindric spiral spring 

 of small angle was given for the first time 

 in 1S4S by Prof. James Thomson, andc he 

 authors follow his method in investigat- 

 ing the laws governing the behaviour of 

 spiral springs generally. They find that 

 if the centres of all cross-sections of the 

 wire, or strip, forming the spring lie on 

 a right circular cylinder of radius r ; i( 

 the spiral have everywhere an inclination 

 a to the plane perpendicular to the axis of 

 the cylinder, and if a force F act at one end 

 of the spring along the axis, the other end 

 of the spring being fixed ; if B is the 

 flexural rigidity of the wire in the oscu- 

 lating plane, and if A is the torsional 

 rigidity about the spiral line at any place ; 

 if the angular motion, in a horizontal 

 plane, of the free end of the spring rela- 

 tively to the fixed end be called <p, and if 

 the axial increase of length be called d, 

 and the whole length of the spring along 

 the spiral /, then — 



(1). 



COv ( 



A 



°(a-b) 



1 sin 2 o\ 



~TT/ 



(2)- 



<p = / F r si 

 and 

 d = I F I s - ( 



Assuming for the general investigation 

 that the cross-section of the wire is elliptic, 

 it is found that the rotation of the free end 

 of a spring like Fig. I or Fig. 2 is greater 

 the greater the inequality in the principal 

 diameters of the elliptic section. 



