692 Prof. Horace Lamb on the 



rotation as an ' error.' Such errors are unavoidable, but we 

 may (still following Helmholtz) inquire under what law 

 regulating the ocular positions the mean square of the errors 

 involved, under similar conditions in different parts of the 

 field, is a minimum. 



Any position of the eyeball may be defined on the usual 

 plan by three angular coordinates. We denote by 6 the angle 

 which the visual axis (OP) makes with its primary position 

 OA, by i^ the inclination of the plane AOP to the horizontal 

 plane through OA, and by <£> the angle which some plane 

 (OPQ) through OP, fixed in the eyeball, makes with the plane 

 AOP. For definiteness this plane OPQ may be taken to be 

 that which in the primary position is horizontal. On 

 Listing's Jaw we should then have $= — i/r, but this is not at 



Fig. 5. 



present assumed. Since the position of the eyeball is 

 determined by the direction of the visual axis, we regard <j> 

 as a function of the independent variables 6 and -v/r, to be 

 ascertained. 



By a known kinematical formula any small displacement 

 involves a rotation whose component about OP is 



d<f)+ cos 6 d^ (1) 



If ds be the displacement of P on the spherical field, making 

 an angle e with the plane of #, we have 



dQ= cos eds, sin 6 dyjr = sin € ds. . . . (2) 



Hence (1) becomes 



{&"" + A(i +C0S ')H^ • • (3) 



The mean square of this for all directions of ds is 



H(sf)"+^(]>«»)>-- • ■ w 



We have to take the mean value of this in the field over 

 which the visual axis can range, for a given standard of ds. 



