438 Transactions of the Society. 



Additional Note. 



Since the reading of this paper Dr. G. Lindsay Johnson has very 

 obligingly drawn my attention to a paper by Prof. J. D. Everett, which 

 appears in the 18th volume of the Proceedings of the Physical Society, 

 p. 166. In that paper Prof. Everett animadverts upon the form of 

 Hockin's proof here discussed, and reproduces the original form from the 

 ■Journal of this Society, 1881, ser. 2, iv.'p. 337. 



Of that original proof it would not be correct to say that Hockin's 

 construction does not satisfy the condition of yielding a plane image of 

 a plane object. On the contrary, his construction is sufficiently general 

 to include both figs. 107 and 108 of the foregoing diagrams, and the 

 mathematics of Hockin's own proof cannot be called in question. But 

 the failure to isolate the case of the plane image of a plane object 

 {fig. 108) is itself a vice. The proof, even in its authentic form, is too 

 general, and by reason of this generality is unnecessarily limited to the 

 case of infinitesimally small images. 



Note II.— The Ultimate Limit of Resolving Power. 



Helmholtz, as is stated above (p. 416), lays down, and without proof, 

 the proposition that adjacent lines in a ruled surface will be indistin- 

 guishable if their diffraction fringes overlap completely, so that the free 

 edge of the one lies upon the engaged edge of the other. This is the 

 same thing as saying that they will be indistinguishable if their adjacent 

 edges lie nearer together than the length of one-half of the diameter of 

 the false disc of the antipoint, for, as we have seen, the antipoint is the 

 image of the diffraction fringe surrounding a luminous point. The 

 difficulty of dealing with this proposition arises from the fact that Helm- 

 holtz does not, in fact, adduce any proof of it, so that there is nothing 

 but his ipse dixit to be met. But perhaps it is not illegitimate to supply 

 this defect from another source, and as Lord Rayleigh has laid down 

 and discussed what appears to be in substance the same proposition in 

 an article on " Resolving or Separating Power of Optical Instruments," * 

 I borrow the following demonstration from that source. To prevent 

 any misconception it should be added that Lord Rayleigh's paper is 

 written with special reference to the spectroscope and that its author 

 is not, so far as I am aware, directly responsible for the proposition 

 that half a wave-length of light is the physical limit of resolving power 

 in the Microscope. But the passage which I propose to quote appears to 

 be equally applicable to both instruments, and in any case furnishes 

 material by the aid of which the case of the Microscope may be very 

 usefully investigated. 



" The curve A B C D represents the values of u~ 2 sin u from u = 

 to u = 3 7r. The part corresponding to negative values of u is similar, 

 A being a line of symmetry. 



* Scientific Papers, Lord Rayleigh, vol. i. 1869-1881, p. 420. 



