the Laws of Chromatic Dispersion, 261 



mental errors ; for an inaccuracy in the determination of the 

 three assumed as the basis of calculation may be of such a cha- 

 racter as greatly to increase the discrepancies between the ob- 

 served and calculated indices of the other lines, and even to 

 make them appear to exceed the legitimate bounds of probable 

 error. Cases, however, may occur in which it may be necessary 

 to rest contented with determining the other five indices from 

 three being given. 



To illustrate this point, and to exhibit the method of pro- 

 cedure in such a case, suppose that for the bisulphide of carbon 

 we have given only ^A, M D, and ^11, and let the corrected in- 

 dices be assumed as those given. The first step is to find the 

 exponent n. Call A n "^A=a„, &+*& =4, and H^H = h n . 

 Make (A n -D M ) -r(D"— H w ) = and (a n -<4)-K4~^) = $ 

 these two quantities, ©and 6, being either integral or fractional. 

 When the proper exponent is correctly found, then is © = 0. If 

 the exponent be too low, <B) is the greater; but if the exponent 

 be too high, 6 is the greater. Now, although the difference 

 between ® and 6 does not vary rateably with the difference be- 

 tween two exponents when the latter are widely apart, yet, if 

 the difference between the two exponents do not exceed 0*5, the 

 corresponding variations in the value of ® — 6, or 6— ®, are so 

 nearly rateable as to suffice for the purpose of calculation. 

 Hence if two exponents be taken, one too high, the other too 

 low, and differing by not more than 0*5, the true exponent may 

 be easily found. 



If the medium be highly dispersive, like the oil of cassia, the 

 two exponents may be 3 and 3*5. If the dispersive power be 

 somewhat less, as in bisulphide of carbon, take 2*5 and 3, and 

 for lower dispersive powers 2 and 2*5, 1*5 and 2, 1 and 1*5, 

 according to a scale which will be easily found by practice. 

 Then by dividing by 5 the variation in the value of the dif- 

 ference between ® and 6, we obtain nearly the rate for each 0*1 

 of exponent, and can thence arrive at the true exponent to 

 within 0*1, or even nearer if required. 



Thus, in the case of bisulphide of carbon, with exponent 2*5 we 

 have® = 1-525176, and = 1-523900, whence@-0=O-OO1276. 

 With exponent 3 we have © = 1-786120 and 6 = 1*791053, 

 whence 6- (8) = 0004933. Adding these two, we obtain 0*006209 

 as the total variation, corresponding to a difference in the expo- 

 nent of 0*5 ; and one-fifth of this amount, or 0*001242, is the 

 variation for each 0*1 of exponent. It is thus seen that a rise 

 of 0*1 from 2*5 will make (B) so nearly equal to 0, that 2'6 may 

 be regarded as the true exponent. 



This quantity found, it is easy to determine log e n and a n% 



