328 Mr Larmor, The application of the Spherometer [May 2, 



ment, that it is in fact the indicatrix plane of that point, and to 

 deduce that the reading gives the mean of the principal curva- 

 tures of the surface; this result is correct, but the assumption just 

 mentioned is erroneous. 



To obtain a rigorous investigation, let us assume that the 

 points of support form an isosceles triangle, let the base subtend 

 an angle 2a at the centre of the circumscribing circle, and let c be 

 the radius of this circle and h the ordinate drawn from its centre 

 up to the surface. If this ordinate is taken as axis of z, the equa- 

 tion of the surface will be 



,2 2 



where (p, q, — 1) is the direction of the tangent plane at the origin, 

 and R , R 2 are the radii of principal curvature. As the three legs 

 rest on the surface, we have 



7 //i x • //i x 1 2 fcos 2 (0 + a) sm 2 (0 + a)) 

 h = cp cos (0 + a) + cq sin (0 + a) + \<? \ K R ' + ^ '-I 



7 //i \ • //i x t o(cos 2 (# — a) sin 2 (0 — a)} 



h = cp cos (0 - a) + cq sin (0 - a) + |c 2 ^ H ^ -\ , 



[ K 1 K 2 ) 



7 n -a .12 f cos2 & sin 2 

 h = — cp cos — cqsm0 + Jc <— ~ 1 — ^ — 



where it + is the azimuth of the vertex of the triangle of sup- 

 port. We are to eliminate p, q, and so connect h with R v R 2 and 

 0. By addition of the^ first pair of relations 



2h = 2cp cos cos a + 2cq sin cos a + 2 c ) ( p~ + 75- ) 



+ (V - »-) cos 2 # cos 2a 

 therefore by use of the third 

 2h (1 + cos a) = -k 2 i(^ + Jj-) (1 + cos a) 



+ (jt- - -=-} cos 20 (cos 2« + cos a)l , 

 or on rejecting the factor 1 -f cos a, 



7 = (i + i) + (1; - s-j cos 2 ^ 2 cos a - !)• 



The value of h therefore depends on the azimuth except in 

 one case, when a is ^ir so that the triangle of support is equi- 

 lateral, which is the case referred to above. The quantity involved 



