232 OF THE PARTS OF [SECT. vn. 



When A B is equal to C D, the point E is in the middle of the length of the 

 bar B D. 



491. RULE i. With any proportion between the lengths of the bars 

 A B and CD; for instance, if A B : C D : : n : m. Then from the number n sub- 

 tract half the square root of four times its square, less one, for a first number. 

 Also from the number m subtract half the square root of four times its square, less 

 one, for a second number. l Divide the first number by the first added to the 

 second, and the quotient multiplied by the length, B D, of the link or bar, will 

 give the distance of the point E from B. 



Example. Let A B be to C D as 2 : 3 ; then 2 x 2 x 4 = 16, and 16 - 1 = 15, 

 of which the square root is 3'873 nearly, and its half is 1*9365 ; and 2 1'9365 

 is -0635 for the first number. Next 3 x 3 x 4 = 36 ; and 36 - 1 = 35, of which 

 the square root is 5-916, and its half is 2'958 ; and 3 - 2'958 = -042, therefore 



0635 

 -0635 + . 042 = : -602 nearly. 



Hence the length of the link or bar B D, multiplied by the decimal '602, is the 

 distance of the point E from B, or B E = -602 B D, when A B is to C D as 2 

 to 3. Or if the point E be given, then B E divided by '602 will give B D, the 

 length of the bar, link, or distance of the point of connexion D from B. The 

 parallel motions of the engines in Plates xi. and xix. are examples. 



The following rule is much easier in its application, and equally good for 

 ordinary purposes. 



RULE ii. Divide the link into two parts, having the same proportion to each 

 other as the beams, placing the greater part next to the shorter beam. 



o 



This rule applied to the preceding example gives B E = a , 3 B D = '600 B D. 



492. SECOND CASE. In this case, to a bar which moves on an axis at A, Fig. 5. 

 Plates x. (A) and (B) conceive three shorter bars to be added to the end, so as 

 to form with a part of the bar the parallelogram B D G F ; and let another bar D C, 

 which moves on a centre at its extremity C, be attached to the lower angle D of 

 the parallelogram which is most distant from the centre C, round which the bar 

 moves. Then the piston rod being attached to the other lower angle G of the 

 parallelogram, its motion will be nearly rectilineal in the direction G H. 



The purpose of rendering the stress in the same direction during the whole of 

 the stroke would determine me to prefer the construction which renders the links 

 B D, F G, vertical at both extremities of the stroke, as in Plate x. (A) : this is not 

 however the usual mode in practice, for the line of motion of the piston rod is 



1 These two numbers are the versed sines of the arcs described by the beam and radius bar, 

 n and m expressing the radii. ED.] 



