686 Prof. Horace Lamb on the 



the interpretation of visual sensations would be enormously 

 complicated. 



The law which defines the position of the eyeball in terms- 

 of the direction of the visual axis is attributed to Listing 

 (1857). It has been tested by various observers and ninv 

 be regarded as established, at all events for normal eyes. 

 There is a certain ' primary ' direction OA of the visual 

 axis, or line of sight, to which all others are referred. 

 Roughly, this may be described as the position assumed when 

 with head erect we look towards a distant point of the 

 horizon, straight in front. For the present purpose a more 

 precise definition is unnecessary. Listing's statement is 

 equivalent to this, that when the visual axis takes any other 

 direction OP the position finally assumed by the eyeball is- 

 that which would be derived from the primary position by 

 a rotation about an axis perpendicular both to OA and OP,. 

 through an angle AOP. In the actual transition from OA 

 to OP, the gaze may wander about in any manner, but the- 

 final position of the eyeball must always be the same, in virtue 

 of Donders' law 7 . 



The various directions of the visual axis may be distin- 

 guished by their intersections with a spherical surface of 

 arbitrary radius described about as centre. This will be 

 referred to as the 'spherical field.' If A represent the 

 primary, and P any secondary position, the above rotation is 

 conveniently indicated, after the manner of Donldn and 

 Hamilton, by the great-circle arc AP. 



Helmholtz investigates in the first instance the relation 

 between any two secondary positions, represented (say) by P 

 and Q on the spherical field. This can be found very 

 simply. It is known that, if ABC be any spherical triangle,, 

 successive rotations of a rigid body represented bv 2BC, 

 2CA, 2AB will restore the body to its initial position, and 

 accordingly that successive rotations 2Bl-, 2CA are equi- 

 valent to 2BA*. It follows, in the present application,, 

 that the transition from one secondary position (P) to 

 another (Q) is equivalent to a rotation 2XY, where X, T are 

 the middle points of the arcs AP, AQ respectively. For the 

 transition may be supposed made, first from P to A ; and 

 then from A to Q. 



* The theorems, in this form, are due to Donkin, Phil. Mag. (3) 

 vol. xxxvi. p. 428 (1850), and (4) vol. i. p. 187 (185J). They were 

 piren independently by Hamilton, ' Lectures on Quaternions/ pp. 328- 

 330 (1853) as direct interpretations of quaternion formulae. Uonldn's 

 simple proof will be familiar to readers of liouth's ' Advanced .Rigid 

 Dynamics.' 



