of the Interference of Homogeneous Light, 91 



and r H = y' 2 +x 2 , r, 2 = y % + d~ l for the points at which the 

 curves intersect ; and also y = r cos <p, y = r\ cos ;£. 



Eliminating cos <f>, cos %, r* and r} by means of these 

 equations, we find 



» (y + *) 2 



It now remains to establish the requisite relation between 

 Q and Q'. For this purpose, putting for the difference of the 

 paths g'y the letter ?/, which we lately found the means of de- 

 termining, letting fall the perpendicular £ 8 upon the axis of 

 the lower pencil, and calling the distance e 8 = 8, of which we 

 easily get the value, 

 we have on the axes of the pencils 



r\ = r; £ -8 + 3/, 



and Q %* = Q' — . - 8 + y 



COS I COS I * 



Q ' = Q +-c&+*-* = Q+C ' 

 by putting C = ~^- +8-3/. 



Returning to our former equations, we have 



Q* = {f + **) ^±^, (Q + C)* * (y + ^) ^±*T; 



we may now eliminate Q, and obtain an equation containing 

 only x,y and constants, and which will represent generally 

 the curve in which the central points of interference should 

 take place. 



Eliminating, we arrive at this equation : 



scy^t^^ — ^ 



v y y 



+ {y+vf- *» {jL f^ - (V +*) 2 - w 



To get quit of the sign of the square root, it is necessary 

 to raise both sides to the square ; and putting for x' its value 

 x— jS, we have 



• o *+]£. ^ c$S = £* (^ - ^ ) 



-, 8/ si2±*)! +^(iLt|)l + (,+*)• _ ( y+»)i-c«Y 



y j y 



It will be seen that the involution to the second power is 



N2 



