31(j PROFESSOR KELLAND ON THE THEORETICAL INVESTIGATION OF 



is received on the screen is exactly the same as if there were no interference at 

 all, but it is differently distributed. 



These conclusions are deduced from the expressions for the intensity of the 

 light at any point of the screen given by the Astronomer Royal in his Tracts. 

 They depend, therefore, on HUYGHENS'S principle, and are proper for the examina- 

 tion of the truth of that principle, or of its more exact statement. It is to this 

 matter, which was only casually touched on in the Memoir referred to, that I wish 

 to direct attention at present. 



The principle, as stated by Mr AIRY, is this : " The effect of any wave in 

 disturbing any given point, may be found by taking the front of the wave at any 

 given time, dividing it into an indefinite number of small parts, considering the 

 agitation of each of these small parts as the cause of a small wave, which will 

 disturb the given point, and finding, by summation or integration, the aggregate 

 of all the disturbances of the given point, produced by the small waves coming 

 from all points of the great wave." Here it is evident that there is introduced as 

 the coefficient of agitation, the area of the front of the wave, or at least of a por- 

 tion of it, so that we must divide again by some area, or the product of some two 

 lines, in order to bring the agitation at one point to depend simply on the agita- 

 tion at another. It will be the object of the following pages to ascertain what 

 that divisor must be. This will be effected by conceiving that it is some constant 

 quantity, which we are led to do from the results of our former investigations, 

 and then from the expression for the total quantity of light received on the screen 

 to deduce its value. We proceed, then, to solve the following problems. 



PROB. I. A series of plane naves passes through a parallelogram, and is trans- 

 mitted to a screen by means of a lens, the focus of which is on its surface ; 

 to find the total quantity of light which is received on the screen. 



Let 2 e, 2f be the length and breadth of the parallelogram,^, q the co-ordinates 

 of a point in the screen, measured from the focus parallel respectively to e and/ 

 b the distance of the lens from the screen. Then, if a be the coefficient of vibra- 

 tion, or a 2 the intensity or quantity of light on a unit of surface of the incident 

 wave ; D the divisor in question ; the intensity at the point whose co-ordinates 

 are p and q is 



16 e* f sin -- sin 



D 2 J \2TTyf l>\ ) \2irpe b\ ' AIRY'S Tracts, p. 324. 



Hence the total intensity, or the quantity of light on the screen is 



sn sn 



