THE TESTING OF REFLECTING SURFACES. 267 
being exposed in succession, while the rest of the surface respec- 
tively remains covered. For each exposed zone the point is 
found along the axis of mirror A, where the knife edge, on being 
moved transversally across the path of the rays, shuts out the 
light suddenly and symmetrically from the whole area of the 
bright ring. This indicates that the knife edge crosses the cone 
of rays, reflected by that ring, exactly at the vertex, and therefore 
determines the centre of curvature of the ring. The amount of 
longitudinal displacement of the knife edge, which is required in 
order to meet in succession the vertex of the cone of rays 
reflected by the other rings, determines the difference in the length 
of the radius of curvature of the successive zones, which displace- 
ment is capable of being accurately measured by a micrometer to 
within + > ths. of an inch, or less. 
In this manner the form of the primary mirror is first deter- 
mined separately ; then the secondary mirror is introduced, as 
shown in diagram, and new readings are taken for the combined 
surfaces. From the data thus obtained, by eliminating the 
observed aberrations for the great mirror, those of the convex 
mirror, and thence its form, can be ascertained. 
These observations are of extreme delicacy, and require to be 
‘carried out in the middle of the night, when everything is quiet, 
and a fairly uniform and constant temperature can be more 
satisfactorily attained throughout the buildings where these tests 
are made. 
It is shown, in conclusion, that with the Cassegrain telescope 
the best definition is obtainable when the form of the primary 
mirror is a paraboloide, and that of the secondary a hyperbolide ; 
that deviations from these forms in the two surfaces when acting 
in combination, will generally tend to compensate each other, and 
will be entirely neutralized when the amount of error in the 
convex mirror is double that of the concave mirror. Consequently 
an error in the concave mirror is doubled by a perfectly figured 
convex mirror; also that an error in the convex mirror is not 
altered in quantity by a perfectly-figured concave mirror, but is by 
the latter changed in kind ; so that, if overcorrected, it would, in 
conjunction with the concave, appear undercorrected, or give in 
combination, the same image as an undercorrected concave surface 
acting singly. 
No. 12.—A GENERAL EXPRESSION FOR FLOW 
IN TUBES. 
By G. H. Kniss, F-R.A:S., LS. 
(Read January 11, 1898.) 
