Screw* 127 



v it will be seen that the part of the force parallel to the axis of the 

 screw which is exerted upon any point of the thread, is to the 

 force which it is necessary to apply at this point to prevent the 

 turning, as the base of the inclined plane is to its height, that is, 

 as the circumference of the circle, which would be described by 

 the point of application, is to the perpendicular distance between 

 two adjacent threads. Therefore, if we call p the first force, and 

 q' the second, and tf the distance of the point of application from 

 the axis, h the height of the supposed inclined plane or perpen- 

 dicular distance between two adjacent threads, and 2 n the ratio 

 of the radius of a circle to its circumference, we shall have 



p : q / :: %Sn : h. 



But each point of the nut is not supported directly; the 

 whole is subjected to a certain power </, applied at some point of 

 the nut whose distance from the axis may be represented by D. 

 It is hence evident, that D being greater than tf, there will be 

 necessary for each point, a force so much the less, according 

 as the distance D is greater ; so that if we call q the part of this 

 force which at the distance D is capable of the same effort as q' 

 is at the distance <?, we shall have 



q' i q :: D : tf. 



Multiplying this proportion by the former, we shall have 

 p : q :: 2nD$ : h$ :: 2xD i h; 



that is, for each point of the nut that rests upon the thread of the 

 screw, there is the same ratio between the force exerted parallel 

 to the axis, and that which at a given distance D prevents the 

 turning ; and this ratio is that of 2 n D to h. Now 2 n D is the 

 circumference of the circle which wouid be described by the 

 power q in turning ; we conclude, therefore, that the sum of all 

 the forces p which urge the nut parallel to the axis, is to the sum 

 of all the powers q necessary to prevent the turning ; as the cir- 

 cumference of the circle which would be described by the pow. 

 er , is to the distance between two adjacent threads of the 

 screw. 



216. Hence the force which it is necessary to employ par- 

 allel to the axis of the screw, to prevent the power q from turning 

 the nut, must be to this power 9, as the circumference which this 



