Motion for the Spectroscope. 103 



C R 6, D S a are constant ; and by the symmetry of tlie figure 

 we see that A P 6? = D S a, and B Q c = C R 6. From the 



equations to the curves we get tan A.V d — ^ — r, and 



51 . . T 



tan B Q c= -^ — r, ; and as it will be best to have the intersec- 

 tions as nearly as possible at right angles, we must make one 

 of these angles exceed a right angle by just as much as the 

 other foils short of one. This gives tan AF d + tan B Q c= 0, 

 or /= >/5. The equations to the curves now become 



6 6 

 a ic. a/5 a/5 a/5 



r-=.ae ''^ , r=.ae 2 ^ r = ae '^ , r=-ae * . 

 From these equations we have 



APjt? = aSs =tan-i4^=24 6 



BQ(7 = iR7' =tan-i^=41 48 



c^q =CR^'=tan-»-^=53 18 



V 



Hence 



APjo = DSs=tan-'-^=60 48 



APrf=DSa =84 54 



BQc=^»RG =95 6 



so that the intersections of the slits in the disks only differ 

 about 5° 6^ from right angles. 



In the model, a was put =25, and the position of points 

 in the fourth curve were calculated from the equation 



^' = 25e~4 ^ by first transforming this equation into the form 



r=log->{l-397940-Jolog-K*385209+ log6/)}, 



and thence obtaining t for every degree in the value of 0. 

 The firstj second, third, and fifth curves were obtained by put- 

 ting J, ^, f , and \ of the value of Q against the same value of 

 r. The limits between which the curves must be drawn will 

 depend on the refractive power of the glass employed for the 

 two ends of the spectrum. In the model the angle between 

 consecutive arms varies between 45° and 54°. 



The following Table gives the calculated points in the series 

 of curves : — 



