A, P. Quinhy on Crank Motion. 3l9 



This, I understand to be a case in wh'ch there is no loss 

 of power. 



Let it next be considered what eifect P would have du- 

 ring its descent through some particular, or assumed space. 

 Take Py=AB, and it will be manifest that the effect of 

 P, during its descent from P to y, will be properly express- 

 ed by PxPy, or PxAB = WxBA. Let it also be con- 

 sidered, that during the descent of P to the point y, the 

 wheel and crank will be made to turn through half a revo- 

 lution ; for, since, by cons. AD : CD : : CD : CG ; and by 

 the property of circles AD : CD.'jG : CG, it follows that 

 iG is=CG; and consequently Gtv—2CD=AB=Fy: 

 and, therefore, if the crank be at A when P shall begin to 

 descend, it will have described the arc ADB, and have ar 

 rived at the lower dead point B, at the time that P shall 

 have arrived at y. 



Assume, now, Ca for one position of the crank ; and 

 suppose a shackle bar, az, having upon its upper end the 

 power P',=P, to stand perpendicular, and rest upon the 

 wrist of the crank at a : — it is proposed to consider the ten- 

 dency that P' would have acting upon the wrist of the nank 

 at a, to produce rotation, (or to give angular motion to the 

 wheel and crank) ; in comparison with the tendency which 

 P has to produce rotation, acting upon the teeth of the 

 wheel, at the point G. Produce za to n ; and draw the 

 sine am ; and it will be plain that the tendency of P', to 

 produce rotation, will be to that of P, as C^, or om, to CG. 



If, therefore, CG be taken to express the tendency of P 

 to produce rotation ; then that of P' to produce rotation, 

 will be properly expressed by am ; the perpendicular dis- 

 tance of the point « from the line ACB. And, in general, 

 the tendency of P' to produce rotation, at any point what- 

 ever of the senjicirrle ADB will be expressed by the per- 

 pendicular distance of that point from the line ACB. 



Hence to determine the mean tendency of P' to produce 

 rotation, (in terms of CG,) during its descent from A to B, 

 in the arc ADB, we must find the mean distance of the 

 semicircle ADB from the line ACB ; which, by Vince's 



1^*1^2 CD^ 



Flux. p. 97, is=-' _-; but, CG was made=^y^; and, 



therefore, the mean tendency of P' to produce rotation, du- 



