320 On Printing Presses and their l^heory. 



SUPPLEMENT. 



As the power gained by different combinations like those 

 referred to in the preceding pages seems to have scarcely 

 attracted the notice of writers on Mechanics, I shall subjoio 

 an investigation of such as are most likely to occur in prac- 

 tice, for the information of those who may be concerned in 

 the invention or improvement of machines which contain 

 such combinations, and to whom it may be sometimes im- 

 portant to know with precision the mechanical advantage 

 they gain by different supposed arrangements of machinery, 

 and the strain to which the different parts are subjected. 



Prop. I. Let CB (Plate III. Fig. l.j be a straight rod, 

 moveable about C, and BA another rod, connected by a 

 joint with CB at B, and with its other extremity A confined 

 to move in the hne CA produced : it is required to deter- 

 mine the power which applied at B, in a direction at right 

 angles to CB, will overcome a given resistance acting on 

 the point A, in the dkection AC. 



The power will be to the resistance, in this, as in all oth- 

 er cases, in the inverse I'atio of the velocities of their re- 

 spective points of application. We have therefore only to 

 investigate, for any given position of B and A, the velocity 

 with which B moves, or the circular arc DB increases, 

 compared with that of A's motion on the line CA. For 

 this purpose, draw the perpendicular BP, putAB=a, C 

 B=&, AC = a?, BP=y, and DB = z. Then x= V^^^'i'-H 



. . — y (\y — y Ay 



^/b^-y2 ; and taking the fluxions, diX=^—j=====-\-—pf-^ — * 



ya2 — y2 ■yj b^ — y^ 



b 



T> A. J ~b(^y , ,. . . dr yyfb^—yi , 



But dz=~7-=== ; hence by division -r- = - — - + 



Vb^-y^ ' '' dz b^a2-y^ 



This expression gives the ratio of the velocities of A and B ; 

 and hence is to unity as the power is to the resistance. But 

 by reinstating the values for which x, y, he. were substitu- 

 ted, it admits of simplification. Reducing the terms to a 

 common denominator and restoring their values, it becomes 



BP AC sin BAG sin ABC sin ABC rpi ^ ■ a 

 = AP* CB==-^BAC- irn-BAC=-^BTC* ^^^' ''' '^'"^ P^^" 

 er !s to the resistance overcome, as the sine of the angle 



