MK.MOUIS OF TllK NATIONAL ACADK.MV OK .SClKNriOS. 



41 



This equation has been proved Cor small values of « only iind would therefore hold ROtKl if 

 eitlier w or tji m were substituted for sin o. In tlie last form, /// <.» replaeinj; sin (.>, the e(|uatioM 

 was lirst given by La','ranv:e and was a most iniiM.rtaiit eontribution t» tin- theory of optics; bnt 

 it is possible to sliow that tlif form which we have tfivcn above is ri;:idly true, indcpcndeiilly of 

 the size of u, provided oidy lliat tlio incident and (inaily iclra.l.d wmv c mh t.iccsuic both splicrical. 

 t. «'., that the system is free from spherical aberration. 



Let pq (Fig. 3) be the incident wave c' and // </' the finally refracted wave surfaces CAfi. If 

 & is limited by a diapbram at </, tlien ha+i is also limited at a point </', which is the cnrrcftponding 

 point to r/; that is to say, the i)oint in a+i where all the light energy comes from a region in & 

 indelinitely near q. Call the semiangular aperture of <•' and of ca+i, oj' and £»'a+i, as indicated in 

 the figure. Now suppose the incident wave to be inclined by an indefinitely small angle o'c\ then 

 tin- finally refracted wave will also be incdined by an infinitely small angle at 7>', which will be 

 equal to «A+i Ca+i. It is apparent from the figure that in the new oblique waves <ii, is a corrnsponit- 

 ing point to q, since the wave surface is propagated in the direction of its normal, and, for the 

 same reason, </,' is a corresponding point to q', hence */]' on the finally refracted obli(iue wave is a 

 corresponding i)oint to </i. But since p and />' are corresponding points for both wave systems, 

 the time required for liglit waves to jiass from q to q' and from r/, to qi' is, in each case, equal to 

 the time reciuired in going from /; to p' ; hence the time rcfjuired for i)rogression from </ to </, is 

 equal to the time fi"om q' to qi'. The velocity of propagation is the last medium, however, is p„ 

 times as great as in the first; consequently we have 



M. • qq\=<i'(i\'- 



Bv inspection of the figure we see that 



qq\=pq • "'<•' cos i o)' 



<l'l\'=P'q'' '>K+\ Ca+1 cos h Ct'A+l. 



Combining these three equations and substituting the trigonometrical expression for the chords 

 pq and p'q', we have — 



n' sin i CO 



4 f./=2 Oa+1 sin h co^+\ cos 4 m^+u 



wlience we derive inunediately the equation (c). 



'Plus highly important relation is essential in calculating the absolute o]itical ])ower of aiiy 

 optical apparatus excei»t the telescope. Its truth was assumed liy I'rol'. Abbe in iiis celebrated 



