TRANSACTIONS OF THE SECTIONS. .16S 



a 



$T _ 0_ ST •] 



Y 6 « Y p I 



, «' ,_ 8T ,_£,__ ST ^ '^ 



^~ y'^ ~ 8«"2^ y^ ~ S/3'' 



The first three coefficients, P P, P', which enter by (5.) into 

 the expression of the term T'^\ are those on which the focal 

 lengths, the magnifying powers, and the chromatic aberrations 

 depend : the spherical aberrations, whether for direct or in- 

 clined rays, from a near or distant object, at either side of the 

 instrument (but not too far from the axis), depend on the six 

 other coefficients, Q Q^ Q' Q^^ Q', Q", in the expression of 

 the term T'*^ Here, then, we have already a new and remark- 

 able pi'operty of object-glasses, and eye-glasses, and other 

 optical instruments of revolution ; namely, that all the circum- 

 stances of their spherical aberrations, however varied by di- 

 stance or inclination, depend (usually) on the values of six 

 RADICAL CONSTANTS OF ABERRATION, and may be deduced from 

 these six numbers by uniform and general processes. And as, 

 by employing general symbols to denote the constant co- 

 efficients or elements of an elliptic orbit, it is possible to deduce 

 j-esults extending to all such orbits, which can afterwards be 

 particularised for each ; so, by employing general symbols for 

 the six constants of spherical aberration, suggested by the 

 foregoing theory, it is possible to deduce general results re- 

 specting the aberrational properties of optical instruments of 

 revolution, and to combine these afterwards with the pecu- 

 liarities of each particular instrument by substituting the nu- 

 merical values of its own particular constants. The author 

 proceeds to mention some of the general consequences to which 

 this view has conducted him, respecting the aberrational pro- 

 perties of optical instruments of this kind. 



When a luminous point is placed on the axis of an object- 

 glass, or eye-glass, or other instrument of revolution, and when 

 its rays are not refracted or reflected so as to converge exactly 

 to, or diverge exactly from, one common focus, they become, 

 as it is well known, all tangents to one caustic surface of revo- 

 lution, and they all intersect the axis, at least when they are 

 prolonged, if necessary, behind the instrument. But if the 

 luminous point be anywhere out of the axis, the arrange- 

 ment of the final rays becomes less simple than before. They 

 are not now all tangents to the meridian of a surface of revolu- 

 tion, nor do they all intersect the axis of the instrument ; they 

 become, by another known theorem, the tangents to hvo causfia 



