ROPE MAKINC. 



HOPE MAKIXd. 



180 



ft. Th yU> root* of unity xit in pairs <>f the form oo f sin f 

 V(-l'. Than pair* are and *-', ' and -, aanda~',a and 

 ^*J 



l*t UM question now be to lad Uw oth rooU of - 1. If we now 

 ukr 



- 1 - co. ( + tt) +dn (w + itw) . V( - 1 ) 



w* h\f all the oth root* in UM dutinrt value* of the formula 





l.rt - - f n - V(- 1). then the 7th root* required are a, a 1 , 



*..... *-', beginning with i= 0. and ending with J~o 1. Tluw, 

 if (i b* any one root, all the odd power* of M (positive or negative) are 

 alio rooU, but do not contain among them all the root* unless the 

 value uf it +1, from which p i derived, be prime to g. Thus if 9 15, 

 and if M . wo have (unc* *= 1) 



^- ^-. -1, ^-o->, ? = & = *, M ==a; 



a that we only get, from the powers of a', the dirtinct root* a', a", 

 1. u>. ,. :1 . which are abo the fifth root* of -1. Rut if 21 + 1 be 

 prime to f, all the oth root* of 1 may be obtained from /i. And 

 if m b any factor of 7 with an odd quotient, all the mth root* of 1 

 are among the oth root*. Also theae </tli roota occur in pairs of the 

 form cos f> J; sin Q\ ( - 1), the pairs being a and a*-', s and a 1 *-' 

 Ac. or o and -' , o> sind a- 1 , Ac. 



Every fth root of 1 ia one of the (2o)th root* of + l,and the 



(2o)th rooU of + 1 consist of all the fth roota of 1 and all the oth 



root* of + 1. 



The following equation* will also be easily proved ; 



Aa it is not our object here to write on the applications of these 

 fonnulw, but only to supply an article of reference for those who may 

 have forgotten or imperfectly learnt the groundwork of this very 

 important branch of analysis, we finish here, referring to SF.IIII - t.,r 

 such applications as fall within the plan of this work. 



HOI'K MAKING. A rope is a combination of fibres of hemp, or 

 other material, so arranged as to form a flexible and tenacious cord or 

 band ; retaining, as far as possible, their collective strength. The 

 name rope is generally confined to the larger descriptions of cordage, 

 mich as exceed an inch in circumference ; though the principles of 

 formation are much the same for cordage of every size. * 



If the fibres used by the ropemaker were of sufficient length, the 

 moat effectual way of obtaining their united strength would be to lay 

 them side by side, fastened together at each end, so as to form a 

 bundle or i-kein ; but, as the fibres of hemp do not, on an average, exceed 



Fig. 1. An.iiy.i. of s Uop. 



the length of 3J fwt, it Incomes necessary, in order to obtain rope of 

 greater length, w to twine them together that the strength of any 

 fibre ahall be insufficient to overcome the resistance caused 

 by the friction of tho e surrounding and compressing it ; so that it 

 will sooner break than be drawn out from the mass. This requi- 



site entanglement is produced by twisting, which causes the fibres to 

 compress each other; and it not only enables .tha ropemaker to 

 produce cordage of any required length, but also, by making the rope 

 hard and compact, increases its durability, and to resist the 



penetration of water, which would rapidly im]..-iir its strength. While 

 however some degree of twist is absolutely essential to the cohesion of 

 a rope, any twist beyond that which simply prevents the fibres being 

 drawn out without breaking, is injurious. A skein of fibres, or a rope, 

 may be twisted so hard that any further attempt at twisting would 

 break it ; and such n skein will evidently have no power to support a 

 weight, each fibre being already strained to the utmost extent that it 

 will bear. In fact, whatever force is exerted by any fibre in compress- 

 ing the rest, may be considered the same as a weight hanging on that 

 fibre, and must be subtracted from its absolute strength before its 

 useful effect can be ascertained ; the available strength of a r..p 

 ' the remainder of the absolute strength of its component fibre 

 deducting the force exerted in twisting them. 



Wrrv a rope to be formed by simply twisting together, in one 

 tion, the whole of the fibres of which it is composed, there would lie 

 nothing to prevent its untwisting as Boon aw left to itself. It i 

 fore necessary to twist the fibres in comparatively small portion 

 so to combine these into a rope that the tendency to untwist in one 

 part may counteract the like tendency in another. Thus the same 

 force which would cause the component parts, if separate, to become 

 loose or untwisted, is employed, when they are combined into a rope, 

 to keep the whole firm and compact. This is illustrated in 

 which may be considered as the unravelled end of a cablet or 

 small cable. The cablet a may be untwisted into three smaller 

 ropes, b; untwisting either of these in the opposite direction, we timl 

 it to consist of three smaller ropes, c ; each of these may be untwisted 

 into several small strings, il ; and each of these consists of several 

 distinct fibres of hemp, e. Thus several fibres form a yarn, several 

 yarns a strand, three strands a rope, and three ropes a cablet. 



With fg. 2 as an illustration, we may briefly describe the hand 



Fig. 2, Rope-yarn Spinning. 



method of rope-making. The first process consists in twisting the 

 hemp into thick threads, called rope-yarns. This process, which 

 les ordinary spinning, is performed with various kinds ,,i 

 machinery. The common mode of spinning rope-yarns by haml 



I in the rope-ground, or rope-walk, an enclosed Blip <>t lex,] 

 ground 600 feet or more in length. As many of th . )S of a 



ropery would l.c impeded l.y wet weather, or by the unchecked I 

 the sun, it is not unusual to cover the walk with a slight roof. At one 

 end of this ground a spinning-wheel is set up, which gives motion l, v a 

 band to several small rollers or whirls. Kach whirl has a small hook 



1 "I on the end of its axis next the walk. Knch of the spin 



provided with a bundle of dressed hemp, laid round his waist, with 

 the bight or double in front, and the ends passing . at his 



Lack. He draws out a sufficient number of fibres to form a rop 

 of the required size ; and, after slightly twisting th. , with 



his fingers, he attaches them to the hook of a whirl. The whirl I.eini- 

 now set in motion by turning the wheel, the skein is twisted into a 

 rope-yarn; the spinner walking backwards down the ropt-w:ii: 

 )>orting the yarn with one hand, which is protected by a wetted piece 

 of coarse cloth or flannel, while with the other he regulates the quan- 

 tity of fibres drawn from the bundle of hemp by the revolution of the 

 yarn. The degree of twist depends on the velocity with which the 

 wheel is turned, combined with the retrograde pace of 1 1 

 Great care is necessary in this operation to make the yarn ..I unitWm 

 thickness and to supply the hemp equally from both sides . 



idle ; because, if a considerable body of hemp be supplied 

 that , becoming too thin, it will not combine perfectly with it but 

 will lorm a loosely connected wrapper ; and any irregularity in the last- 

 mentioned particular will cause the fibres to bear the strain unequally 



