214 



UNDULATORY THEOEY OF LIGHT. 



instead of their sines, 

 comes — 



Let these chords be r and r' . 



Then the equation be- 



a^ — (T 

 which, if V — v' be made constant, is the equation of 

 a lemniscaic. The annexed figure represents a Icm- 

 niscate, or curve whose distinguishing property is 

 that the products of every pair of radii vcctores, 

 drawn from two poLar points, and intersecting in the 

 curve, arc eq\ial to each other, and to a constant 

 quantity. If PQ, the distance between the poles, 

 be bisected at A. and PA made = q, then the con- 

 stant vahie of Pllx QR, divided by q, is called the 



]>ara7nctcr, and may be represented by j>. Put Prt = r, QR = ?-', AN = x, and 



RN = y. The construction gives, immediately — 



r''^z=[qj^x)^-\.7f; r-={q — x)'^-\-')/i whence 



Or rh-'-'^{q^+a?+iff—^q^x~=/q^. [43.] 



In the case in hand, the parameter is the quotient found by dividing the 

 second member of the equation for the velocities, in its last form, by q. The 

 value of q itself may Ibe directly measured, if the 

 chromatic image be thrown upon a screen, as was 

 done by Sir John Herschel in his study of the forms 

 of these curves ; or it may be assumed at pleasure, 

 from a knoAvledge of the angle between the axes. 

 Thus, if AB(JD be the lamina, and aa' , hh' the axes, 

 then, to the eye at E the poles are a and b in direc- 

 tions parallel to aa' and bb' ; and half their distance 

 is the value of q. The rings, however, may be re- 

 ferred to any distance, as EP; and the poles will then 

 be at Q and R The distance EP and the angle REQ are all that is necessary 

 to determine q, which is now PQ. It must be observed, howevei-, that for a 

 projection on this scale, the value of the constant, or second member of the 

 conation above, must be increased in the ratio of the square of the distance of 

 QR to that of AB from the eye. 



When the direction in which the rays reach the eye is such that the differ- 

 ence of path of the two rays is half an undulation, tlaere will be seen, in homo- 

 geneous light — the analyzer being crossed upon the polarizer — the first bright 

 rin"-. When the difference becomes an entire undulation, the first dai'k ring 

 will appear. The parameter of the lemniscate changes with every new ring. 

 For the bright rings, the parameters will evidently form an arithmetical series, 

 correspondmg to the odd numbers 1, 3, 5, 7, &:c. For the dark rings there will 

 be a similar series of values, proportional to the even numbers. 



The Icmniscates are not perfect, (though some of them are nearly so,) because 

 Ave have admitted some small errors into our assumption. The inner curves 

 also will, in many cases, form ellipses around a single pole. It is obvious that 

 this must be the case when the constant is less than q^. For q^ is the smallest 

 value that the product of the radii vectores can have; and when the parameter 

 is not equal to q, there can be no lemniscate. 



When the analyzer is crossed upon the polarizer, in observing these curves, 

 if the plane of the axes is in the plane of polarization of the incident light, 

 there will be seen a black cross intersecting the system symmetrically; the 

 principal bar of which will coincide with the plane of the axes. The transverse 

 bar will pass at right angles to this, half way between the poles. In these two 



