384 Transactions of the Society. 



upon one of the plane wave-fronts. Now, it is a fundamental 

 assumption upon which the undulatory theory of light rests, that 

 every point upon a wave-front radiates light in all directions, as if 

 it were itself a primary source of light. Therefore, the fact that 

 any such point radiates light along the zero axis is proof that it 

 must also radiate light along the inclined axis now in question. 

 If then we have light coming off from every point of this polyphase 

 surface parallel to an axis, at right angles to the surface itself, 

 we may infer that that light will be propagated undiminished in 

 that direction, for between rays of light that move in parallel paths 

 diere can be no interference. We shall therefore have plane poly- 

 phase fronts passing the aperture A A', just as plane wave-fronts 

 pass the aperture A A, refracted by similar appliances to the 

 focal point F', and producing there upon another index particle a 

 disturbance proportioned to — proportioned to what? Not now 

 proportioned to the area of the polyphasal front that passes the 

 aperture, for the various phases of any given front, when they are 

 made by focussing coincident in space and time, will not reinforce 

 one another. On the contrary, the impulse which started from the 

 point of intersection shown in the diagram with the plane wave- 

 front 12 will be exactly equal and opposite to the impulse which 

 started from the point which the same polyphase front had in 

 common with the wave-front 10. Therefore, these two impulses 

 will cancel one another. It is thus evident that there are ineffec- 

 tive elements in these polyphase fronts, and that their light-yielding 

 power can only be estimated by making the necessary deduction 

 on this account. In the case of the particular series of fronts now 

 in question, it will be at once apparent that the necessary deduction 

 leaves nothing over. For just as the two points 10 and 12 

 paired off and cancelled one another, every other point will have 

 its pair also. Take for example any point adjacent to 10 and 

 call it 10a. There must be another point similarly situated with 

 regard to 12, which we may call 12a, that mil become in the 

 focus the pair and equivalent of 10a, cancelling its effect. Thus 

 the algebraical sum of all the impulses received from any one of 

 these series of polyphase fronts will be 2 = 0, a fact which is 

 indicated on the diagram by showing the convergent beam and 

 focus in full black. 



It will be useful to pursue this line of investigation a step 

 farther, but more convenient to use another diagram than to 

 accumulate more details upon fig. 76. Fig. 77 reproduces the 

 essential parts for this purpose of fig. 76 and will be understood 

 without further description. 



Here we have to consider first, the polyphasal surface <f> 3 which 

 contains not only one complete set of phases as does (f> 2 , but also one 

 half-set over, that is to say, it contains three half-sets of undula- 

 tion phases. It is clear, therefore, that as in the case last discussed, 



