48 



MECHANICS. 



P x C = x x D 



and by the properties of the wheel and 

 axle established in (50), 



x x R = W x r 

 Multiplying these equalities, we have 



pxCxajxR = o?xDxWxr" 

 Omitting the common multiplier x, we 



have 



PxCxR = WxDxr 



or P : W : : D x r : C x R ; 



that is, " the power is to the weight as 

 the distance between the threads multi- 

 plied by- the radius of the axle to the 

 circumference Described by the power, 

 multiplied by the radius of the wheel." 

 . The condition of equilibrium of this 

 machine has been mistaken by some 

 writers, and the error seems to have 

 crept from one treatise to another. 

 Instead of the circumference described 

 by the power, the radius is used which 

 gives the machine only about a sixth of 

 its true efficacy. 



(102.) The very slow motion which 

 may be imparted to a screw by a very 

 considerable motion in the power, ren- 

 ders it an instrument peculiarly well 

 adapted to the measurement of very 

 minute spaces. The manner of apply- 

 ing it to this purpose is easily explained. 

 Suppose that a screw is cut so as to 

 have iifty threads in an inch, and that 

 round its head is placed a graduated 

 circle, on which an index, attached L to 

 the screw, plays. In one revolution of 

 the screw its point, or anything moved 

 by its point, is moved through a space 

 equal to the fiftieth part of an inch. 

 The circle on which the index plays 

 rmiy be easily divided into 100 equal 

 parts, and it follows that the motion of 

 the index through one of these parts 

 corresponds to one-hundreth part of a 

 complete revolution : since, in a com- 

 plete revolution, the screw moves 

 through the fiftieth part of an inch, it 

 follows, that when the index moves over 

 one division of the circle, the screw 

 moves through the five-thousandth part 

 of an inch. 



A screw constructed for this purpose 

 is called a micrometer screw : it is used 

 with great effect in astronomical instru- 

 ments, where very minute portions of 

 degrees or divisions on graduated in- 

 struments are to be ascertained. The 

 limit of accuracy of any divided instru- 

 ment adapted for measuring spaces or 

 distances is primarily the magnitude of 

 the smallest division on it. Kit be re- 

 quired to determine the distance from 



any given division to a point which is 

 placed somewhere between two divi- 

 sions, it is easy to conclude that the 

 distance sought is greater than a certain 

 number of divisions, and less than 

 a number greater than that by one. 

 But how much greater than the one or 

 less than the other, the mere gradua- 

 tion of the instrument does not indicate. 

 Now, suppose that a micrometer screw 

 is placed on the instrument, its length 

 being parallel to the graduated face, 

 and that the point of the screw, or ra- 

 ther, a wire which is moved by the 

 point of the screw, is brought exactly 

 opposite to one of those divisions be- 

 tween which the point, whose exact po- 

 sition is to be determined, lies. If the 

 screw be turned until the wire is moved 

 by its point from coincidence with the 

 adjacent division till it coincides with 

 the point, the number of turns of the 

 screw, and parts of a turn, will indicate 

 exactly the distance of the point from 

 the adjacent division. 



We may give an example of the ap- 

 plication of the screw to this purpose 

 in the steel-yard (44). If the loop 

 which bears the sliding weight P carries 

 inserted in it a micrometer screw, the 

 point of which is adjusted so as to 

 mark the place on the graduated arm 

 G B, at which the weight P is to be 

 considered as acting ; and suppose the 

 screw is such, that in sixteen turns its 

 point would move over one division of 

 the arm, which we will suppose gradu- 

 ated for pounds, let us suppose that 

 when the weight W is counterpoised, 

 the point of the screw is between the 

 tenth and eleventh division of the arm 

 G B. It is evident, then, that the 

 weight is more than ten pounds, and 

 less than eleven pounds. Let the screw 

 be turned until its point moves from 

 the intermediate position to the tenth - 

 division, and note the number "of turns 

 suppose it seven : that would be equi- 

 valent to seven-sixteenths of a division, 

 or seven-sixteenths of a pound, that is, 

 seven ounces. The weight is, therefore, 

 ten pounds seven ounces. In like man- 

 ner, if there had been but 5J turns, the 

 weight would be 10 pounds 5| ounces, 

 and so on. 



Hunter's screw is peculiarly well 

 adapted to micrometrical purposes, be- 

 cause it gives an indefinitely slow mo- 

 tion, without requiring a very exquisite- 

 ly fine thread, which the simple screw 

 \vould require in this case. 



