46G 
LIEUT.-COL. A. E. CLAEKE ON STANDAEDS OF LENGTH. 
curved surface of contact of the second cylinder is in its axis a b, and suppose r to he 
the radius of the spherical surface, R being the radius of the corresponding spherical 
terminal surface of the cylinder A B of the bar P. 
Suppose, in the first case, the bars to be so adjusted that the axis a b is in the same 
line with the axis A B. In this case the length of the horizontal projection of the line 
joining the centres of the spherical surfaces is 
=R+r- 
JU2 
2 1 
t** 
ll + r 
f • 
where e is the distance of the centre of the terminal spherical surface of the cylinder 
above its axis A B, or the distance from A B of that point of the terminal surface at 
which the normal to that surface is horizontal. But if the cylinder a b, still maintaining 
contact, were raised by the quantity e, the common tangent plane at the point of contact 
would be vertical, and the distance of the centres of the spherical surfaces would be 
r-j-R. Hence the har is apparently shortened by the quantity 
2 r + R 
Secondly , for the case of an actual contact in the centre of the terminal disk of the 
cylinder of P or in the axis A B, let the axis a b be below A B by the quantity 
re 
E’ 
In this case the contact is in the actual centre of the terminal disk of the bar P, and 
the horizontal projection of the line joining the centres is 
so that the error in this case is 
t> i i B + r 
li + r—f.-jpj-c-; 
a 
i> • 1>2 
e\ 
To reduce these to numerical results for the bar P, the value of R is 2-00 inches, 
e—-£v inch. For the contact-apparatus the radius r=0 - 75 + - 03 inch, obtained, as in 
the case of the bar P, by the measurement of optical images formed by reflection. 
Expressed in millionths of a yard, one inch=27778; hence, since the same defect exists 
at each end of P : — 
Case 1. When the axes of the cylinders are at the same height, 
e 2 
Correction =— -w=6*31 ; 
r + E 
Case 2. When the contact is at the centre of the disk of P, 
Correction = ^ =11*93. 
The correction in the latter case is nearly double that in the former, while the actual 
difference in position of the points of contact in the two cases is but of an inch. 
