Interference-Bands. 203 



width is proportional to X, the width for the nth order is 

 a different function dependent upon a. It is with the latter 

 that we are concerned when, by means of the prism, the 

 nth bars have been brought to coincidence. 



If the glasses be in contact, as is usually supposed in the 

 theory of Newton's rings, a = ; and therefore, by (41), 

 S# oc \*, or the width of the band of the nth order varies as 

 the square root of the wave-length, instead of as the first 

 power. Even in this case the overlapping and subsequent 

 obliteration of the bands is much retarded by the use of the 

 prism ; but the full development of the phenomenon demands 

 that a should be finite. Let us inquire what is the condition 

 in order that the width of the band of the nth order may be 

 stationary, as X varies. By (41) it is necessary that the 

 variation of X 2 /(JnX — a) should vanish. Hence 



2X(±nX-a) -in\ 2 = 0, 

 ° r a = ±n\ (42) 



The thickness of the plate where the nth band for X is 

 formed being ^nX, equation (42) may be taken as signifying 

 that the thickness must be half due to curvature and half to 

 imperfect contact at the place of nearest approach. If this 

 condition be satisfied, the achromatism of the nth band, 

 effected by the prism, carries with it the achromatism of a 

 large number of neighbouring bands*. 



We will return presently to the consideration of the 

 spherically curved glasses used by Newton, and to the 

 explanation of some of the phenomena which he observed ; 

 but in the meantime it will be convenient to state the theory 

 of straight bands in a more analytical form. 



Analytical Statement. 



If the coordinate £ represent the situation of the nth band, 

 of wave-length X, then, in any case of straight bands, j-f may 

 be regarded as a function of n and X, or, conversely, n (not 

 necessarily integral) may be regarded as a function of £ 

 and X. If we write 



» = *«,*), («) 



and expand by Taylor's theorem, 



+ |g 2 (8^ + ..., .... (44) 

 * Enc. Brit, Ware-Theory, p. 428 (1888). 



