justification is made in the case of propagation in three dimensions 

 by classical reasoning due to Kirchoff and discussed notably by 

 Bateman (3), on pages 184.-186, as well as by numerous analytical 

 treatises (10) with respect to t he propagation of waves. It re- 

 lates the value of the propagation function at a point P to the 

 values which the function and its normal derivative take on a surface 

 around the point P„ Generally one applies the formula obtained to 

 diffraction by means of a certain number of hypotheses which are de- 

 tailed by DeBroglie (9) on page 91 and following „ However, for 

 vraves in three dimensions the auxiliary function that must be in- 

 troduced in addition to the propagation function has a simple ex- 

 pression, it is rigorously /r, where r is the distance from a 

 point source of movement. P to a point M situated on the surface 

 surrounding point p„ It is not the same in a harmonic movement in 

 two dimensions, where the expression of movement originating in 

 a point source involves Bessel functions of zero order of the first 

 and second types (see Lamb (2) pages 296-29J, and 527-529) for 

 which the asmytotic expression occurs in r~2 and applies only to a 

 distance from point P which is greater than 2 or 3 wave lengths „ 

 The first term neglected in the alternate series development 

 of the exact movement is r~3/~ c However, in employing the exact 

 expression of the movement starting at a point source P one arrives 

 easily at an expression analogous to that obtained by Kirchoff, 

 but which applies to a cylindrical movement and more precisely to 

 a wave or swell. The expression gives the movement at point P 

 if one knows the velocity potential and its normal derivative on 

 a cylinder with normal generatrices in the plane of still water <> 

 One deduces from this formula a Huyghen ! s principle which applies 

 to oblique incidence, but which, containing the asmytotic ex- 

 pression of elementary movement and not the exact expression, in- 

 volves therefore an approximation and is practically applicable 

 onlv if the distance PM is greater than 2 or 3 wave lengths „ 



Referring to Figure 1, the letters CC define a wave crest, 

 let S be the cross section of the cylinder on which one assumes 

 the propagation function and its normal derivative are known „ 

 Let cc be the angle between the wave crest and the tangent to 

 S, oc is the angle of incidence^ let 6 be the angle between the 

 normal to the crest and the straight line W , and r the distance 

 from the point P where one seeks to determine the movement, to a 

 very small elementary source, situated on So The depth is supposed 

 constant throughout <> 



The transposition of the ?archoff formula, which transposes 

 in a fashion the roles of P and M permits the establishment of 

 a Huyghen's principle of the following form The movement at 

 P is obtained always in first approximation by considering that 

 the movement is the resultant of elementary movements of the 

 sources, such as M, distributed on the surface S (known movement), 



15 



