THE MODERN REFLECTING TELESCOPE. 31 



plane (at right angles to the axis) in which the pinhole itself lies. This is the 

 simplest formula which can be used, but it is not the most useful in practice. 



In testing pai'aboloids at the centei' of cui-vature the writei' has always used 

 the following method and formula: The illuminated pinhole remains fixed at the 

 center of cuivatui-e of the central parts of the mirror, i. e., at a distance 2 F from 

 the vertex, where F is the focal length. The intervals, measured along the axis, 

 between the reflected foci of the various zones, are now twice as great as those 

 given by the method described in the preceding paragraph; consequently these 

 foci can now be detei'mined with twice the accuracy which can be attained by that 

 method. Only the rays i-eflected from the parts of the paraboloid very near to 

 the vei-tex are now brought to a focus in the plane of the pinhole. If the para- 

 boloidal figui-e is perfect, the rays reflected from any very naiTOw zone whose semi- 

 diameter is R are now brought to a focus at a distance -^ + — =^back of the plane 



2 r 16 F^ ^ 



the pinhole, i. e., at a distance 2 F + -~^ -\- from the vertex of the paraboloid. 



2 r lor'' 



The quantity ^^3 is so small in the case of mirrors of moderate size and of ordinary 



I'atios of apertui'e to focal length that it can be neglected ; even in testing the 

 outermost zones of the 5-foot mirror of 25 feet focal length, this quantity is less 



R2 



than 0.002 inch, while the (quantity —^amounts to 1^ inches. 



Now let us consider what is the best method of determining the planes of the 

 reflected foci. Draper, Common, and other workers used an eyepiece for this purpose ; 

 this serves well for mirroi's of moderate angular apei'ture, but for mirrors in which the 

 ratio of aperture to focal length is as great as 1 to 5 or 1 to 6 this method presents 

 serious difficulties; if narrow zones are used the image in the eyepiece is blurred 

 and indistinct on account of the diffraction effect produced by the edges of the 

 zonal openings in tlie diaphragm, while if wide zones are used the difference of 

 focus of the inner and outer parts of a zone is so great that the image shows 

 evidence of marked aberration ; with neither narrow nor wide zones can the 

 position of the focus be determined with very great accuracy. 



In Publicatians of the A. S. P., vol. xiv.. No. 87, Hussey gives a foi'mula for 

 the position of the " cii'cle of least confusion " when a zone of given width is used ; 

 if Hussey's formula were employed and the pinhole were made very small and 

 i-ound, with smooth edges, it is probable that inuoh greater accuracy could be 

 attained than by the use of an eye})iece in the oi'dinaiy way. 



The method of locating the reflected foci which is used by the writer is as 

 follows; it is capable of surprising accuracj'^ when the optician has become experi- 

 enced in its use. The reflected focus of a zone is found with the knife-edge, pre- 

 cisely as the focus of a spherical miri'or is found. The knife-edge is moved across 

 the reflected cone from the left ; if the left side of \\w zone is seen to darken first, 

 the knifeedcre is inside of the focus; if the riorht side darkens first, the knife-edjje 

 is outside of the focus; when the riirht and left sides of the zone darken simul- 



