On Printing Pres-^es and their Theory. 325 



Draw BP perpendicular to DE and join AP. Then the 

 plane ACDP is perpendicular to the plane BEB'; and BP 

 drawn perpendicular to their line of common section is also 

 perpendicular to the plane ACDP, and therefore to the line 

 PA which it meets in that plane. Hence APB is a right 

 angled triangle, and AP^ + PB ' = AB = . If BD be put = 

 r, AC = r', AB = a, CD = x, and BE=:z, BP will be - 

 come==sin 2r,DPr=:cos ^, and AP» (^DC^'+AC^PD^) 

 = x^-\-{r'^ cos zf. Hence sin'* r + a?2 + (rVcos z)^__a2. 

 Expanding (r' -r cos z)^, and substituting r~ for sin ^^-f-cos" 

 z, we have a^^+r^— 2 r' cos z-\-r^ =a^. Taking the flux- 

 ions, 2 xdx = 2 r' d (cos z) = —^l2 sin z 6z ; and by resolu- 



tion, dx : —dz :: - sin z : x. But 6x and —dz express the 

 velocities of the points to which the weight and the power are 

 respectively applied ; so that power : weight : : - sin r : a^ : : 



sin BDE to rad. AC : CD. 



Cor. I. When BDE = or 180°, sin BDE vanishes, and 

 the gain of power becomes infinite. But DE is evidently 

 the position which BB' assumes when AB and A'B' come 

 into the same vertical plane. Hence the weight infinitely 

 exceeds the power necessary to support it, when the two 

 rods come into the same vertical plane. When the angle 

 EDB is so small, or the line DC so large, that the variation 

 of DC may be neglected, the power necessary to support 

 a given weight will vary as the sine of EDB, the angular 

 distance from the position at which it becomes evanescent. 



Cor 2. Every thing else being the same, the gain of pow- 

 er from this combination will increase in the same propor- 

 tion as the distance of the lowest points of the rods from C 

 is diminished. 



Cor. 3. If, as will generally be the case, the two extrem- 

 ities of each rod are equidistant from the central line CF, 

 or AC=:::BD, the power will be to the weight simply a? 

 BP : CD. 



