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LXXT. The Kinematics of tlie Eye. 

 By Horace Lamb, F.R.S.* 



THIS subject should be o£ interest to mathematicians, 

 for it is almost the only field where the theory of 

 Finite Rotations, so elegant geometrically, finds a practical 

 -application. 



The theory of the movements of the eye was investigated 

 very completely by Helmholtz f, and forms a considerable 

 chapter of his Physiol ogische Optik %. The mathematical 

 part of the exposition is rather elaborate, and has I think 

 somewhat obscured the essential simplicity of the matter. 

 There is a decided gain in this respect, as I wish to show, 

 if we adopt a purely geometrical treatment, based mainly 

 on theorems of Donkin and Hamilton, which were indeed 

 anterior in date to Helmholtz's work but could hardly have 

 been known to him. 



A brief recapitulation of the physiological principles 

 involved may be convenient. Fortunately these are simple, 

 and (I believe) well established. 



The movements considered are of course relative to the 

 hpad, which for the present purpose is to be regarded as 

 fixed. The precise attitude of the head is unimportant, but 

 for facility of statement it is supposed to be erect. The eye- 

 ball rotates very approximately about a fixed point §, and 

 so far as the muscular equipment is concerned it has the 

 usual three degrees of freedom. But in its normal operation 

 the movements are so coordinated that there is virtually 

 freedom only of the second order, the position of the eye as 

 ;a whole being completely determined by the direction of the 

 4 visual axis ||/ i. e. of the line drawn from to that point of 

 the external field which is the object of direct vision. This 

 is 'Donders' Law' (1847). The limitation to freedom of 

 movement which it asserts is essential in order that the same 

 object, seen in the same position relative to the head, should 

 affect the same elements of the retina whenever the gaze is 

 directed to the same point of it. If this were not the case 



* Communicated by the Author. 



t " Ueber die normalen Bewegungen des menscliliehen Auges " 

 (1863), reprinted in Wissenschaftliche Abhandlungen, vol. ii. p. 800. 



X lsted. 18G6; Landed. 1896. 



§ Here and elsewhere in the present subject the conditions are 

 slightly idealized. 



|| German ' Blicklinie? For reasons connected with the structure of 

 the e^-e this is not exactly coincident with the line drawn to the object 

 from the anterior nodal point. The distinction is hardly important except 

 in the case of very near objects. 



Phil. Mag. S. 6. Vol. 38. No. 228. Dec, 1919. 3 B 



