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SUMMARY OF CURRENT RESEARCHES RELATING TO 
New Method for Constructing and Calculating the Place, Position, 
and Size of Images formed by Lenses or Compound Optical Systems.’ 
— The late Prof. G. Govi wrote : — “ The theory of lenses and of compound 
systems has taken a new form, and reached far greater perfection since 
Moebius, Gauss, Listing, and others have introduced the consideration 
of certain planes and cardinal points, which simplify the construction 
of the place, position, and size of the images, allowing account to be 
taken of the thickness of the refractive medium traversed by the light. 
But the preparatory operations, either as constructions or calculations, 
by which we succeed in determining the place of the points and cardinal 
planes in lenses or systems, are long and wearisome, and often out of 
proportion to the importance of the result we hope to obtain ; and, 
above all, it is always most difficult to determine by experiment the place 
of these planes and points in lenses already worked or in optical systems 
already constructed. 
Physicists, therefore, in spite of the practical methods and instru- 
ments proposed for the purpose by Cornu, Gariel, and others, are for the 
most part limited to considering the lens as having no thickness, and to 
calculate directly and for every limiting surface the path taken by the 
rays in traversing the given media, thus sacrificing a part (and at 
times not a small one) of the necessary precision, or increasing the 
fatigue of the calculations when many determinations of the same optical 
system are in question. 
The suggestion, therefore, of a quicker method for constructing and 
calculating the images given by thick lenses will not be unwelcome to 
students, the same method being also applicable to any optical system 
whatever. 
This method requires the determination of two points which, very 
probably, have not until now been taken into consideration by physicists 
or mathematicians who have treated of these matters ; probably they 
passed them unawares, because if any one had pointed out their impor- 
tance and usefulness they would at once have been recognized, and the 
very latest treatises on optics would have recalled them. 
The two' new points, by which the theory of lenses is very much 
simplified, and which are easily determined by observation, are the 
images of the centres of curvature of the two faces, anterior and posterior, 
of the lens, seen through that one of the two faces to which they do not 
belong. In order to obtain them it is necessary to suppose that the 
luminous rays diverging from the centre of curvature of one face, or 
converging towards it, meet the second face of the lens where by 
refraction they are made to converge towards the image of this same 
centre, or diverge from that image when it becomes virtual. We 
thus have on the axis of the lens the places of the two images q and q lt 
(fig. 9), of the centres c and c 1} and of the curvature of the two faces al 
and 6Zj. 
Having fixed the position of these two points, which we may call the 
centric points of the given lenticular system, nothing else is needed to 
determine any conjugate focus of a point situated upon the axis or out- 
side the principal axis of the system, and to obtain the size and position 
Rend. R. Accad. Lincei, iv. (1888) pp. 655-60 (3 figs.). 
