18.92.] to Surfaces which are not Spherical. 329 



in the formula is then -rr+ t> ; and by referring back to the 



original case of a spherical surface we see that the instrument 

 measures the arithmetic mean of the principal curvatures. 



Thus for example the equilateral form of the instrument may 

 be conveniently used to measure the curvature of a cylindrical 

 lens or a cylindrical pipe, but for that purpose its indication must 

 be doubled. 



The equilateral form will be of no use for testing deviation 

 from sphericity at a given point of a surface. The isosceles form 

 may however be so used, the difference of the extreme curvature- 

 indications given by it for any point being by the above formula 



(srs:) (2cosa ~ 1) ' 



that is directly proportional to the difference of the principal 

 curvatures. The curvature may thus be completely explored*. 



In all these formulae the usual assumption is made that the 

 span of the instrument is small compared with the radii of curva- 

 ture of the surface. 



If the instrument had four legs at the corners of a rectangle, 

 there would be only two positions in azimuth, corresponding to the 

 sections of greatest and least curvature, in which it would rest 

 firmly at a given point on a surface, with all its legs in contact ; 

 and the plane of contact would in this case be parallel to the 

 tangent plane at the summit of the surface. The readings for 

 these positions would give 



cos 2 a sin 2 a. 



. sin 2 a cos 2 a 



and ~R~ + ~~R~ ' 



where a is an angle made by a diagonal of the rectangle with a 

 side ; so that the values of both principal curvatures might thus 

 be determined. 



* Mr H. F. Newall informs me that an isosceles spherometer is used by 

 Dr Common for exploring the curvatures of his large specula. 



