436 



Transactions of the Society. 



are two plane wave-fronts at an angle to one another as shown ; 

 c x and c 2 are the two points in the principal focal plane in which they 

 respectively come to focus ; v\ ... i\ x is the spherical image of c . . . e lf 

 and rj . . . rj 2 is the spherical image of «... e 2 . In this figure four 

 optical paths are traced, and our author first shows that they are all 

 equal. This is easily done, for they are paths connecting conjugate points 

 on two wave-fronts and their images. They must therefore at least 

 form pairs of equal paths, for the two rays from the wave-front e . . . e x 

 to its focus c x must be optically equal to one another, as also the two 

 rays from its image 77 ... ^ to the same focus. Therefore the total 

 path € ... 77 = ej ... Vi- 



Fig. 107. 



In like manner, it can be shown that 



€.,. 



V-i 



But since c x . . . vi and « 2 • • • V2 are both equal to e ... 77, they 

 must be equal to one another. Now, tracing these two paths on the 

 diagram, we observe that the part from e x to 7i 2 is common to them, and 

 deducting this common part we are left with e 2 . . . e 2 at one end, and 

 772 . . . t) x at the other end, which therefore must also be equal to one 

 another. 



We have therefore the two triangles € e x c 2 and 77 ^ ?i 2 , so proportioned 

 that the perpendicular c x . . . e 2 of the one is equal to the perpendicular 

 ri l . . .77 2 of the other. As shown in the diagram, vv 1 v< i is a curvilinear 

 triangle, but if we assume that the wave-fronts are made extremely 

 narrow, it will approximate in the end indefinitely to a rectilinear 

 triangle, its perpendicular rj 1 . . . v 2 always remaining equal to e x . . . e 2 

 and its angle at the point 17 being equal to the divergence angle u . In 



like manner, the angle at c of the triangle e ^ e 2 is equal to the divergence 

 angle w g . So much being premised, we now observe that 



sin n v (v • • ' Vi) — sm u e ( € 



h), 



but subject to the proviso that the magnitude yj . . . rj x is so small that 

 although it actually has the form of an arc, that arc may be identified 



with its own tangent 



