^.GG 



UNDULATORY THEORY OF LIGHT, 



component systems of undulations to be first generated bj independent vibra- 

 tions and then combined; or suppose the two vibrations to be first combined, 

 and then to generate a single resultant system of undulations — the resultant 

 system is the same in both cases. On the first hypothesis, we allow two forces 

 to work out their effects separately and then unite the effects; on the second, 

 we unite the forces themselves, and make them unite their effects from the start. 

 Referring to the equation just cited, we see that the resultant molecular ve- 

 locity, when the movements are in the same plane, takes every value according 

 to the difference of phase of the components, from the sum to the di2"erencc of 

 the two component velocities. Thus, if be put =^ 0°, the equation 



Azzi-^ a^+a''^ + 2aa' cosO, will become A:=z ^/ a^ + a''^ + 2aa' i 

 Or, if a~a', AznSo. 

 If ^ = 90° A= ^/a'^+a'^- or, if a — a', A = aV2. 



:a-\-a'. 



If ^=rl80O, A— >/a^+a''^—2ua': 

 If ^ = 60°, A= VoHVH^m' 

 If ^ = 120°, A= Va2 -}-«'-— 



-a — a' \ or, if a=^a', A^O. 

 or, if a^=a', A=a V3. 

 a'; or, if a=^a', A = a. 



It may aid in obt<aining clear conceptions of this subject to employ a graphic 

 illustration. Such notions are very desirable at this point of our progress, if 

 we would understand the application of the theory of undulation to the expla- 

 nation of optical phenomena ; and especially of those of highest interest. In 

 the annexed figure, let the two curves PHA, QMN, represent two undula- 

 tions, whose molecular velocities are the ordinates drawn to the common axis, 

 MNAC, and whose maxima velocities are PP', QQ'. The undulation PHA 

 is the more advanced in position ; but, referred to any common intersecting 

 line as LA, the undulation QMN is the more advanced in phase. 



Figr 37. 



Now we have seen that the resultant maximum molecular velocity, -when 

 these undulations are combined, will be the diagonal of a parallelogram, oi 

 which PP' and QQ/ are the sides, and of which the angle of inclination of 

 the adjacent sides shall be equal to the difference of phase between the compo- 

 nents. Accordingly, from the point A, where the curve representing the undu- 

 lation least advanced in phase crosses the axis, measure off' AC, in the direction 

 of progress, equal to QQ', the maximum velocity of the wave most advanced 

 in phase. From C, measure backward, OB, equal to PP', the other compo- 

 nent maximum velocity. Prom the centre C, with the radius CA, describe 



