2 The Astronomer Royal on the 



ary of the wave, // the distance of the point from the circular 

 plate, c the distance of the point from the normal line. From 

 the point draw a normal to the plate : the distance of the foot 

 of this normal from the centre of the plate will evidently = c. 

 Take the foot of the normal as the origin of polar coordinates 

 for the front of the wave : suppose the whole front of the wave 

 divided into truncated sectors by radii drawn from the foot of 

 the normal : let the angles made by two of these with the line 

 joining the centre of the plate with the foot of the normal be 

 6 and 9 + 86: and let the small truncated sector included be- 

 tween these be divided into small parts by arcs of circles de- 

 scribed with the origin as centre: let the radii of two of these 

 arcs be r and ?• + 8 r. The area included between these two 

 arcs is ultimately = r89 x 8 ?• = 8fi x r 8 r, and the distance 

 of this area from the point in question is ultimately = \/{h^-\-r^)' 

 If then the amplitude of the vibration which it excites at the 

 point in question be a function of its distance, but independent 

 or nearly independent of the angle which the line of distance 

 makes with the front of the wave, the amplitude of the vibra- 

 tion produced by the small area 8fl x rlr will be represented 

 by 86 X 4> ( 'v//i- + r"") X r 8 r, 



and the phase being — \vt — v' Jf + r* k the absolute dis- 

 turbance in the ether at the point in question, produced by 

 the small area 8 6 x rlr, at the time t, will be represented by 



X <?) {Vh^+ r-) X ?• 8 r X sin '^\'vt- ^h^ + r« V 

 = sin ^^ X 8 9 X <p ( ^FT^) . cos ^ \^¥T^. r 8 r 



A. A. 



-cos^4^^ X 86 X <p(^F+P). sin^Vf^+r^.rBr. 



A, A/ 



And these expressions are now to be integrated with respect 

 to r ; then the limits of r, defined by the intersection of the 

 eccentric radius with the circumferences of the two circles 

 (and therefore functions of 6) are to be substituted, and then 

 the whole is to be integrated with respect to 9. 



The expressions involving ?• cannot be integrated in a finite 

 form so long as the general symbol is retained for the func- 

 tion. If, however, we expand it in the form 



f 1 a/a2 I ^2 __ Ji 



^ |/i + {s^h^ + r' - h)j- = 4> (A) + (;,'(/i)_^-il 1* 



^ 1.2 



they may always be integrated. 



86 



