232 
OF THE PARTS OF 
[sect. VII. 
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. 1 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 2x2x4= 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+^042 = ' 602 nearf y- 
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. 
3 
This rule applied to the preceding example gives B E = g + -<yB 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.]