406 



NATURE 



\August 23, 1888 



may be replaced by its non-vertical asymptote, and then it follows 

 from (u) that 



-Px^R _ c } f T _ Cl «K'R\ 

 m x / 



f + 



v . . (13a) 



lm x I 



the right-hand expression consisting of the first two terms of 

 (12a). Wheny = 2, or the molecule consists of a core and two 

 shells, equation (8) becomes 



f = t ~±y t 2 _ a!£ / K i 2R i • k,*r 9 



1 



/w x \ K t 2 



T 8 K. 



- TV 



:i4) 



y — a + 0x + 



yx- 



+ 



8x 2 



Kj 2 - x 1Q - x 



where 



x, y, a, 0, y, have the same meanings as before, and 

 8 = - c 1 2 'R 2 llm 1 . The curve is therefore of the third order with 

 two vertical asymptotes, x ss K x , and x = K 2 , and a third given 

 by the equation 



y = a - yYLf - 8K 2 2 + (£ - 7 - S)x . . . (15) 



If the curve nearly coincides with its asymptotes, 1* 2 will be 

 given approximately in terms of T 2 by (15), except near the 

 critical periods, and as before there will be three cases, viz. : — 



(a) /8 — y — 8 > o, /u. increases as T increases. 

 {b) $ — y - 8 = o, /j. approximately constant. 

 (V) £ - y — 8 < o, fx diminishes as T increases. 



Near the critical periods /x 2 will always diminish as T 

 increases. 



When the condition (a) is fulfilled, and the curve does not 

 approximately coincide with its asymptotes, /j, may continue to 

 decrease as T increases throughout the whole branch of the 

 curve between the two vertical asymptotes, the curve running 

 from the upper left-hand to the lower right-hand side. 



The expansions in powers of T will be different for the three 

 branches, viz. : — 



For T < K tt 



8 \ , _J y 8 



H~ = a + px + x- 



For T > K 2 , 



/u 2 = a - 7K X 2 - 8K 2 2 + 



f &j + nK7 + Kv 



+ &c. 



(16) 



- y - 5) x 



- 1 ( 7 Ki 8 + SK„ 6 ) + &c 

 x l 



For K x < T < K 2 , 



(J - a - 7K X 2 + (j8 - 7) x - 

 8x 3 7I 



(7V + SK 2 4 ) 

 . . . . (16a) 



7^2 



8x 2 



K 2 



7 K/ 



+ 



Kx< 



+ &c. 



(16*) 



The first terms of {16a) are identical with the right-hand side 

 of (15), and therefore if the curve nearly coincides with its 

 asymptotes, it will closely approximate to the curve (14), except 

 near the critical periods. This explains why Cauchy's expansion 

 of (j? in descending powers of T, or of X, gives approximately 

 correct results. In this expansion the coefficient of T 2 vanishes 

 if the asymptote is parallel to the axis of x, viz. if .3 = 7 + 8, 

 or if 



m x = ^(IVRi + K 2 2 R 2 ) (17) 



If 8 = o it reduces to the preceding case ; the curve breaking 

 up into the asymptote x = K 2 ' 2 , and a hyperbola. If 7 = o it 

 breaks up into the asymptote x = K. x 2 and a hyperbola. 



In general, with a greater number of critical periods, if the 

 curve is of the order n, it will have n — 1 vertical, and one other 

 asymptote. To the left of the first vertical asymptote and to the 

 right of the last there will be a hyperbolic branch, and between 

 every two of them will be a branch of the curve proceeding from 

 the upper left-hand to the lower right-hand side, either falling 

 continuously or reaching a minimum, then rising to a maximum, 

 and again falling and approaching the next asymptote. There 

 will be n distinct expansions for /r in powers of T 2 , one for each 

 branch of the curve. In many cases the curve, except near the 

 critical periods, will approximately coincide with its non-vertical 

 asymptote, and there will then ,be the three cases, (a), [b), (c), 

 to consider, as in the previous examples. 



§ 3. — Dispersion and Reflection. 

 It is well known that the spectrum of light of a given kind 

 depends on the function of T 2 serving" to express <**. The 



dispersion in a refracting medium will be designated as normal 

 when, except near the critical periods, /r diminishes without 

 limit as T 2 increases, and anomalous when /j.' 2 increases without 

 limit, or passes through a series of maxima and minima. In 

 the first case the colours of the spectrum will appear in their 

 " natural " order, the smaller values of T 2 corresponding to the 

 blue, and the larger values to the red end of the spectrum. In 

 the examples considered in § 2 the dispersion will accordingly 

 be normal in case (c), and anomalous in case (d), while in case 

 (l>) the spectrum will be compressed into a line. 



When the dispersion is anomalous throughout, the colours 

 will appear in the inverse of the natural order, but it will be 

 otherwise when it is alternately normal and anomalous. 



Consider, for example, the non-vertical asymptote in case (c). 

 Then if there are only two critical periods there will be to the 

 left of the asymptote x = K x 2 , a hyperbolic branch, along 

 which ju 2 will decrease continuously, giving normal dispersion at 

 the blue end of the spectrum above the axis of x. Below this 

 axis jjt will be negative, and therefore /j. will be imaginary, so 

 that light of the corresponding period wili be entirely reflected 

 by the medium. From the point of intersection of the branch 

 of the curve with the axis of x to the point x = Kj 2 there will 

 therefore' be a dark space or absorption band. To the right of 

 this point /u 2 will again decrease from positive infinity to a 

 minimum. 



Suppose this to be at a position for which x = p above the 

 axis of x, the curve will then rise to a maximum, say for .r = q. 

 For p < T 2 < q the light will, then be more strongly refracted 

 than for T 2 <; /, and therefore the corresponding colours will 

 be displaced, and may overlap the colours for which T 2 < p. 

 There will therefore be a dark band at the part of the spectrum 

 which should be occupied by them, but this is not now an 

 absorption band, and may be made to disappear by further 

 dispersion. For T 2 <; q the dispersion will be normal up to 

 the intersection of the branch with the axis of x, from which a 

 dark band will extend to the point x = K„-, after which the 

 dispersion will again become normal. 



Phenomena of this kind have been observed by Kundt and 

 others, and the fact that they follow from the formulae was 

 considered by Thomson to afford important confirmation of the 

 theory. In fact, taking T proportional to A, the preceding 

 equations do not differ essentially from those obtained from 

 quite different phenomena by Sellmayer, von Helmholtz, 

 Lommel, and Ketteler, and which have been shown to be in 

 complete accordance with experiment. 1 



Sir William Thomson, in his Baltimore lectures, came to the 

 conclusion that according to his theory metallic reflection would 

 necessarily cause dispersion. This would be the case if there 

 were only a single expansion for fi 1 , but in the case of most of 

 the metals there are so many lines, distributed over the whole 

 spectrum, that there is no reason for selecting any one in 

 preference to the others. The fact that all the colours are 

 reflected to practically the same extent, which means that jir 

 must be a negative constant, may be completely explained by 

 the assumptions that the corresponding curve of the wth order 

 approximates very closely to its ;/ asymptotes, and that the 

 single non-vertical asymptote is very nearly parallel to the axis 

 p — o. The essential portion of the curve may then be replaced 

 by its horizontal asymptote, as in the cases previously con- 

 sidered, in which J3 - 7 and - 7 - 8 respectively were 

 assumed to be nearly zero. The non-existence of dispersion 

 does not therefore afford an objection to the theory. 



It is easy to see that by a suitable choice of the disposable 

 constants, the curve may be made to practically coincide with 

 its asymptotes, for consider the curve of the third order given 

 by (14). This may be written in the form 



(K-f-x) (K 2 2 -x) (y-a-Px) = yx\Kf - x) + 8x-(K x 2 - x) : 



or 



(K^ - x) (K 2 2 -x)(y-a- px + 7K X 2 + 8K 2 2 + yx + Sx) 



= x 3 (7 + 8 - 7K; 2 - 8K 2 2 ) - x (7V + 8K 2 4 ) 



+ KfK 2 3 (7K 1 2 + 8K 2 2 ), 



and it is evident that when K : 2 and K 2 2 are given, the right- 

 hand member may be made to vanish by taking 7 and 8 small 

 enough, and the required condition will then be fulfilled, since 

 the left-hand member equated to zero represents the three 

 asymptotes. 



1 See Wiillner, " Experimental- Physik," vol. ii. pp. io5 and 169, fourth 

 edition. An outline of the various theories of reflection and refraction 

 will b^ f jund in the British Association Reports for i335 and 1S87. 



