APPLIED MECHANICS. 



[MICROMETER ELLIPTICAL OKAR. 



If we examine any part of the table, as, for instance, 

 the numbers given for U turns to the inch, we finl the 

 following role obtains : The pitch in the first column u 

 final to twice the product of the nmiiliura in the tlnni 

 ami fifth, divided by Uie product of those in the second 



80 X 70 

 and fourth. Thus 14-2 X 35^-55- Again, where no 



compound intermediate is employed, the pitch is twice 

 the number in the fifth column divided by that in the 



second. Thus 6 pitch- 2 X -jQtoe&i. 



MICROMETER. For machinery by which straight 

 lines or the circumferences of circles are divided into a 

 number of equal parts, as for the marking out of scales 

 f jr measuring or astronomical instruments, or for cutting 

 teeth in racks and wheels, the screw and the worm ana 

 wheel are often employed iu order to give the means of 

 delicate subdivision. When a screw is used for such a 

 purpose, it is generally called a micrometer (small 

 measurer) screw, and the principle on which it acts may 

 be thus described : If we suppose an accurately cut 

 screw, having ^th of an inch pitch, to be fitted with a 

 nut, the motion of a screw through one revolution would 

 advance the nut through T ' e th of an inch. If on one 

 end of the screw there were fitted a wheel having its cir- 

 cumference divided into 100 equal parts, the screw might 

 be turned round any number of revolutions or hun- 

 dredths of a revolution, as marked by a fixed index. 

 pointing to the divisions on the wheel. But for every 

 hundredth part of a revolution of the screw, the nut 

 would be advanced yfoth of ^th, tua > is loW th l> &ri o{ 

 an inch ; and by using a screw of still finer pitch, and 

 having a wheel mounted upon it divided into still 

 mailer and more numerous parts, the longitudinal ad- 

 vance of the nut through still smaller fractions of an 

 inch could be effected and estimated. It is by such an 

 arrangement that the fine divisions of mathematical in- 

 struments are traced, and the fine lines traced upon 

 medallion drawings are engraved. For circular division, 

 the screw, instead of moving a nut longitudinally, acts 

 as a worm on the teeth of a wheel, and causes it to 

 revolve through any required part of its circumference. 

 Thus, if we had a worm-wheel with 360 teeth, and a 

 worm with a wheel fitted on its axis, having 3(50 divi- 

 sions, we could move the worm-wheel through gfojth of 

 xfath, that is, TTW tn * a resolution, or any number 

 of such fractions of a revolution. And farther, if the 

 wheel upon the screw were a toothed wheel, and we h:id 

 numerous other toothed wheels, with various numbers 

 of teeth like those we have described for a screw-cutting 

 lathe, we might, by the proper selection of wheels gear- 

 ing with that on the screw, effect the division of the 

 circle into any number of required equal parts. Such 

 apparatus are employed for dividing the circumferences 

 of astronomical instruments, and also for cutting the 

 teeth of wheels. When the number of teeth to be cut 

 is a multiple of some simple numbers, such as the num- 

 ber 60, which U a multiple of some of the numbers 2, 3, 

 4, 6, 6, 10, 12, 16, 20, 30, it is generally easy to select 

 wheels which, in connection with the screw, shall give 

 the required number of equal divisions ; but when the 

 number of teeth U what is called a prime number, such 

 M 69 or 61, which is not capable of subdivision, we must 

 either provide wheels having such a number of teeth 

 already cnt in them, or resort to methods of approxima- 

 tion for their subdivision. Thus, if with the 360 teeth 

 on the worm, and 300 divisions on the screw-wheel, we 

 desired to divide a circle into 61 equal parts, we should 



129 600 

 for each part turn the screw through -'-; 2124J 



Since the total number of divisions due to a complete re- 



volution are 129,600, 

 and since 2124 J X 61- 129,594 }, 



the revolution of the wheel would want ."._ .li\ i~i..m 



of being complete, that is, 



or about 



part of its circumference : a quantity quite inappreciable. 

 I l.LIPTICAL GEAR. Iu several machines it is 

 necessary to provide a rotary motion more rapid at one 

 portion of a revolution than at another, as, for instance, 

 in machines for slotting and planing iron. The iron to 

 be planed, or the tool which planes it, makes a rectilinear 

 stroke slowly in the one direction while the metal is 

 being cut, but may be drawn back rapidly in the op- 

 posite direction when no work is done. For producing 

 motion of this kind, elliptical geared wheels are some- 

 times employed (Fig. 262). The ellipse is a curve, of 



fig- JOJ. 



dirisaona, or ^jJj^-B rerol 324J dir. nearly. Tlie 

 at the end of the process would be found thus : 



\ 



which the line A B is called the greater axis, and each of 

 the two points C and D in that line is called a focus. If 

 to any point E in the circumference of the ellipse, lines 

 C E and D E be drawn from the foci, the sum of their 

 lengths is equal to that of the axis A B. Farther, if the 

 lines C E and D E be prolonged beyond the curve, and 

 a line F G drawn so as to divide equally either of the 

 angles formed by their intersection, the line F G is a 

 nt to the curve at E ; that is, it touches it, but does 

 ut it ; or every part of it, except merely the point 

 E, lies entirely outside of the curve. If, in the pro- 

 longed portions of the lines C E and D E, lengths E K 

 and E H be measured off equal to D E and C E re- 

 spectively, H and K mify be taken as the foci of another 

 ellipse, precisely equal in every respect to the original 

 ellipse, and having the axis L M, and the tangent F G 

 tourhing if at E ; and as F G touches both ellipses, they 

 touch each other at the same point E ; and the length of 

 the line C K, which is made up of C K and E K, or its 

 e.jn.-tl E 1), is equal to A B or L M. 



These peculiar properties of the ellipse enable us to 

 employ elliptical wheels, each revolving round a focu* .-is 

 a centre, C for the one and K for the other, the two f ( i 

 maintaining a constant distance apart, whatever be the 

 relative position in which the wheels lie with respect to 

 each other. The circumferences are cut into teeth and 

 spaces like ordinary circular gearing, the elliptical curves 

 being their pitch lines, on which the equal divisions are 

 set out. 



It will be observed that, during a revolution of these 

 wheels, the point M gears with B, and the point L gears 

 with A ; but as the radii of these points are B and 

 K M, and K L and C A respertively, the angular veto- 



