276 Prof. L. Natanson on the Molecular Theory 



to the axis, and let 7 represent the angle between the phine 

 (pz) and (.vz), a fixed plane through the axis Using r to 

 denote as before the distance between M and V, we have 



[Ef]=N^oi '\dypdpE?> = TSdzo\ *\ dyrdrE?. (1) 



Jo Jo Jo J ~_ Zo \ 



! z~ z I stands here for +(z — z ), the plus or minus sign 

 being taken according as ~ — z is positive or negative. The 

 axes of the vibrators are parallel to the direction of x so that, 



by§i, 



Ef=-B + r*A (2) 



From (1) and (2) we deduce 



[Ef] = 2^^0(1,-1,, (3) 



where 



I A = ±y r drfl- (Z ~J o)2 \A; . . . (4a^ 



j z — zo <- J 



I B = | rdrB (46) 



In calculating I A and Ib we must use in (4), instead of A 

 and B, the expressions (5) and (6), § 2. A difficulty now 

 emerges because, unless some further conditions are assigned, 

 the integrals (4) are really ambiguous. This difficulty is 

 not peculiar to the present discussion ; it arises in a similar 

 manner, for instance, in the Theory of Diffraction. The 

 ambiguity is usually avoided by supposing* that the inte- 

 grated term at the upper limit may be taken to vanish. [An 

 excellent proof of the legitimacy of this mode of procedure 

 will be found in Dr. Fr. Reiche's paper cited above.] On 

 this understanding the values of Ia and Ib are at once 

 obtainable. We find 



r «wA (2 W- a I Z ~ Z « \ - hz *+<A , ,r\ 



J A= - —. - (OJ 



tan | z — z j 



ank w e in ^- a \ z ~ z °\ ~ b *o+ql 



I B = . r p— -\l + ian \ z~z \ }. (6) 



tan I z — z I 



Ooing back to (3) we obtain [E^. 2) ] in the form 



[Ef ] = - 2iran dz NA< 2) e l < t ~ a I *-*« I -**o+g]. . . (7) 

 It may be easily verified that [Ef ] =0 and [Ei 2) ]=0. 

 * Cf. Lord Rayleigh's Scientific Papers, vol. iii. pp. 76-77. 



