Let SF1R and RPQ be corresponding lines ; then RPQ will be parallel to AA since it must meet AA 'at infinity' ; and since to a normal plane corresponds a normal plane the locus of R will be a plane normal to the axis and intersecting it in U. Corresponding to this will be a normal plane intersecting AA in U and it is clear that these same two planes will be obtained by proceeding from the second region to the first region .sx The four points , , U and U determine the transformation completely and it is to be noticed that while U and U are corresponding points and do not correspond .sx Let T be any point which without loss in generality may be taken upon ; to find the corresponding point draw TV parallel to AA meeting the normal plane through U in V and let V be the corresponding point upon the normal plane through U. Let intersect the line PQ in T ; then T will correspond to T. Draw TW and TW perpendicular to AA ; then W and W will be corresponding points .sx Moreover , so that .sx Let N and N be two points upon the axis AA such that and ; then N and N are corresponding points .sx Let any line through N intersect the normal plane through U in M and let NM be the corresponding line intersecting the normal plane through U in M. Then UM = UM and NU= NU , so that NM and NM are parallel .sx Thus corresponding to any straight line through N there will be a straight line through N and these two straight lines will be parallel .sx 2 .sx The transformation considered in the previous paragraph is uniquely determined when and are given in position and also one of the pairs of points U , U and N , N ; these six points are named therefore 'cardinal points' and the normal planes through them 'cardinal planes .sx ' and are called 'principal foci,' U and U 'unit points' and N and N 'nodal' points .sx The following relations hold , therefore :sx and ; .sx and and are named respectively the `first' and `second focal lengths .sx ' No relation between and can be obtained from pure geometry and recourse must be had to a law of optics which states that `the ratio of to depends only upon the optical properties of the two regions in which they are measured' :sx this will be proved subsequently .sx In the figure of 1 let us write , and , so that m is the `magnification' or the ratio of the `image' WT to the object WT ; then from the relations given there we have , so that .sx This result is due to Newton and it gives the point upon the axis in either region corresponding to a given point upon the axis in the other region and also the magnification associated with any pair of conjugate points ; it is seen that the origins of co-ordinates are the principal foci and that the directions of measurement are the same .sx 3 .sx Before considering formulae applicable to particular systems it may be as well to obtain a few more general results .sx Thus let and be the principal foci and U and U the unit points of a symmetrical optical system and let P , P and Q , Q be pairs of corresponding axial points associated respectively with magnifications m and n ; and let x , x and y , y be the co-ordinates of these points referred to the principal foci as origins ; we intend to change these origins to the conjugate points Q and Q , referred to which let the co-ordinates of P and P be u and v respectively .sx from the preceding paragraph ; so that .sx .sx If Q and Q coincide with the unit points , n = 1 , and then .sx ; .sx if f and f be equal we have .sx , while if f and f become infinite while their ratio remains finite according to the relation .sx , and being , so far , undefined constants , we have .sx .sx This relation is of use in 'telescopic' systems .sx The longitudinal magnification at any point upon the axis is proportional therefore to the square of the transverse magnification associated with the point ; and we may find similarly the oblique magnification corresponding to any oblique angle .sx 4 .sx Again let PT be any line through P , intersecting the first unit plane in T and cutting the axis at an angle ; let PT be the corresponding line intersecting the second unit plane at T and cutting the axis at an angle V. Then and .sx , i.e. , i.e. .sx If l and l be the lengths of two corresponding normal lines at P and P we have and then ( 2 ) takes the form .sx , i.e. , this quantity is unaltered by the transformation .sx If the angles involved be small we have .sx or if we assume that .sx Let now the line through P intersect the normal plane through Q in R where QR =y ; and let the corresponding line through P intersect the normal plane through Q in R where QR = y. Then = = from ( 1 ) and this is a constant quantity for rays through R. This may be written .sx , a symmetrical relation .sx It may be shewn that for lines not meeting the axis of the system their projections upon a plane passing through the axis will satisfy this relation .sx The preceding results have all been obtained from a purely geometrical theory of 'collineation' - shewing how much may be derived from the mere notion of 'images' ; no optical principle has been introduced so far .sx We shall see later that the condition that rays from a point near to P and upon the normal plane through P should be brought accurately to a focus at a point near to P and upon the normal plane through P is .sx , .sx and this will be derived from optical principles .sx This is clearly inconsistent with equation ( 2 ) and we draw the conclusion that a 'perfect' optical system is an impossibility .sx 5 .sx Refraction at a Single Spherical Surface .sx The preceding paragraphs have dealt with a purely geometrical transformation and we have now to make an optical application .sx It will be assumed for the purposes of this .sx tract that light is propagated under the form of a wave motion and that the disturbance originating at a point of a homogeneous medium is to be found at a subsequent time upon a concentric sphere the radius of the sphere being proportional to the time interval and depending also upon the nature of the medium .sx Let P be a point upon a spherical wave-front which touches at A the spherical boundary , centre C , between two media B and B ; let a normal at P to the wave surface cut AC produced in Q and let the angle PQA be small ; let this normal cut the bounding surface in S and draw NP and MS normal to AC intersecting AC in N and M respectively .sx After time t the disturbance from P will have reached S and that from A will have reached L , upon AC ; draw the sphere with centre on AC and passing through L and S. It will be shewn that to the first approximation this sphere is the new wave-front .sx Let U and R be the curvatures of the incident wave-front and of the bounding surface respectively , so that .sx 6 .sx If Q be the centre of the new wave-front it will be seen that between the points Q and Q there is a one-to-one correspondence ; moreover the line of propagation PS of the incident disturbance has been transformed into the new line SQ of propagation of the disturbance ; we may employ therefore the geometrical transformation considered in the previous paragraphs .sx It will be observed that the results obtained are legitimate only if we neglect small quantities ; the transformation given therefore is true only as a first approximation and in order to allow for this we must consider the 'aberrations' of the optical system which accordingly will be effected in subsequent chapters .sx A special case of 5 arises when the incident disturbance is reflected at the bounding surface AS as indeed will always be the case partially .sx But here we may write and remember that the disturbance is to be regarded as travelling positively after incidence in the direction CA .sx From we have , so that is a constant of the medium varying indeed inversely as the .sx velocity of the luminous disturbance in the medium ; and it may be made definite by writing , where is the velocity of light in vacuo .sx A particularly simple application of ( 3 ) 5 may be considered .sx Consider two spherical surfaces placed close together and touching at A and let the medium between them be defined by the constant the outer media being the same and defined by the constant unity .sx Then a double application of ( 3 ) 5 and an obvious notation leads to and ; where and are the powers of the surfaces .sx But since the surfaces touch and therefore .sx Thus the effect of a `thin lens' is merely to make a constant addition to the curvature of the incident wave-front .sx 7 .sx Various Formulae .sx The formulae ( 2 ) and ( 3 ) 5 are the fundamental formulae for refraction at a single spherical surface ; we may notice some other forms of these however which will be of use subsequently .sx In the first place we may change the origin of co-ordinates from the pole A to the centre of curvature C ; and it is easily verified that the result is .sx Here J , defined by the last relation , may be named the 'modified power' of the surface ; further the expression will be called the 'reduced' distance , i.e. , it is the geometrical distance multiplied by the constant of the medium in which the distance is measured .sx The physical meaning of this product will be examined subsequently ; ( 1 ) now becomes , if we agree to regard all distances as reduced .sx Here the constant does not appear explicitly and this will be found to be true generally if the modified power be used instead of the ordinary power .sx Again formulae involving angles are frequently of use and ( 2 ) 5 may be modified to this end .sx Thus if SQ and SQ cut the axis AQQ at angles and respectively ( fig. 5 ) we have and approximately .sx Substituting we have , i.e. , .sx 8 .sx The General System the K-Formulae .sx The symmetrical optical system will be composed , in general , of coaxial spherical surfaces separating various media ; let us consider the optical properties of such a system .sx Let n coaxial spherical surfaces , 1 , .sx .. n , separate media whose optical constants are , and let , .sx .. , be the powers of the surfaces ; let , .sx .. be the poles of the surfaces , i.e. , the points of intersection with the axis of the system .sx Let a ray inclined at an angle with the axis in the first medium meet the first surface in a point at distance from the axis ; for the corresponding ray in the second medium let these quantities be and respectively ; and so on .sx Let P be the intersection of the ray with the axis , that of the ray , and so on ; let the separations of the surfaces be , , .sx .. so that , e.g. , .sx Then we have the following relations ; .sx , , , ; .sx where is the 'equivalent' distance AP and .sx From this it is evident that is the numerator of the 2th convergent to the continued fraction while is the numerator of the th convergent to this continued fraction .sx We may denote by the numerator to the last convergent to the continued fraction .sx