i 9 o HERMANN VON HELMHOLTZ 



exactly determine the course of the two processes. Since the 

 four constants a, b, m, and n can only be determined from 

 the results of the experiments, and the values / and 5 are 

 constantly changing, it is certain that within narrow limits of 

 the values / and 5, the above equations must be correct. 

 Whether they hold good for wider limits, or whether the 

 four quantities here given as constants are really independent 

 of 5 and f t can at present be determined only by experi- 

 ment, i.e. comparison of the results of our equations with 

 experience. 



Taking the intensity i of the light that impinges on the eye 

 during the observed time as a constant, and forming the uni- 

 versal integrals of the two linear differential equations for / 

 and 5 with constant co-efficients, Helmholtz finds the value 

 to which the magnitude of excitation approximates increasingly 

 with protracted illumination i t and thence deduces the maximal 

 value of the persistent illumination of the eye as the quantity 



-. If /'and 5 be taken as the limits to which the magnitude 

 n 



of fatigue or of excitation approximates gradually with long 

 illumination t, it follows from the given integral functions that, 

 if 5 and / are both at the outset larger or smaller than the 

 values F and S, which they finally arrive at, 5 must at first 

 exceed or fall short of S, then reach a maximum or minimum, 

 and finally rise or fall once more to the value S, while / con- 

 stantly rises or falls to the value F; and the same is true 

 whether at the outset s>S,f<F or s<S,f>F. 



The first case explains directly 'the alternation of posi- 

 tive and negative after-images, when the eye looks steadily 

 at a constantly illuminated field, or even at the retinal 

 field illuminated with its intrinsic light, on which a lighter 

 or darker object has become temporarily visible, and then 

 disappeared. In the former case excitation and fatigue are 

 simultaneously augmented at the point on the retina that is 

 covered by the image of this object ; in the latter they are 

 simultaneously diminished. Upon the disappearance of the 

 object, excitation and fatigue of the spots involved return 

 gradually to their final value, at which the other parts of the 

 retina have remained. The positive after-image corresponds 

 With the period at which the excitation has not yet reached 



