Pa=( 



1863. 279 [Briggs. 



Let P = the force expended on the arm of the wrench. 



a = the length of arm of wrench. 



a. = the angle of inclination of thread or the developed inclined 

 plane of the screw. 



/?r= one half the angle of the thread itself. 



<p = the coefficient of friction on the thread surface. 



^' = the " " " nut " 



D = diameter of nut outside. 



d== " " bolt. 



dj = " " root of thread. 



Q := load on screw parallel to axis, or in other words, the strain on 

 the bolt, thus : 



( rtan d =b 9P v 1 + Cos^ a. tan^ /J-i j i rX)3-d3"l f 



I Ll zp ^ tan d N/1 + Cos^d. tail^jT" + ^ 'S'Ld^tJ ) 



the first member of the cocflacient of Q being derived from the in- 

 clined plane and the friction of the thread, and the second part from 

 the friction of the nut on its seat. 



This cumbrous equation, by having inserted in it definite values for 

 ^ and f ' and for /3, and the values given by the general formula 

 functions of ad-fc before alluded to, underwent the most astonishing 

 reduction to the form 



Pa = Q(Ad + C). 

 where A = a coefficient and C a constant. 



Of course, A and C have values difi"ering with different values of 

 <p, tp' and /? and also changed by the zp ± terms for screwing and un- 

 screwing. But as, in practice, the value of angle 

 /3 is fixed at 30° 

 (p is fixed at about 0.1 

 ip' " " 0.15 



and the value of a in the first formula is established at 0.096 

 " " cat 0.026 .-. 



_ 1 



^~ 0l)96~d + 0^026 

 and the value of D = 1.734 d+0.1445 

 it results that the formula for summing up becomes 



^-^= 0.164 d +0.008 

 Q 



and for unscrewino- 



o 



Pa 



— - = 0.133 d — a constant so small that it can be neglected. 

 ^{ 

 The investigation went further into the whole consideration of 



