SHEWING IT TO BELONG TO PHYSICAL OPTICS. 149 



then X = u cos a, x' = u cos a, 



y = u sin a, y = u sin a, 



substituting it'' - 2 {a. cos a + h. sin a) u +a^ + b'^ = R'; 



.-. u = a.cosa + b.sm a±\/ R^ -a^ — b" + {a. cos a + i.sin a)' 



= a cos« + i sma±\R-'^±^^:^^^^±±^^ nearly. 

 Similarly u' = a' cos a + 6' sin « ± \r' - «" + ^" - (»' e^ " + ^'' sin a)'| . 



.-. 2^p' = ft' - M = («' — «) COS a + {b' — b) sin a ± (E' - R + &c.) 



To establish the condition that the two branches are in contact at 

 the cusp g, and remembering at the same time that, in an observation, 

 the rain-drop is at a great distance, and that therefore Ji and 7?' are 

 large quantities compared with a, b, a', b', we liave 



when a = 0, u' — u = 0, 



and, a' — a±(R' - R) = 0, 



substituting w' — u = {a — a) cos a + {b' - b) sin a — («' — a) nearly. 



Or, since A' lies on the side of the negative x ; therefore a is negative : 

 also b > b', and they are also both negative, in our problem ; therefore 



ti —u = {a — a') ( 1 — cos a) — {h - b') sin a 



= 2 (a — a') sin^ - — (b- b') sin a. 



To put this into a form for use in calculation, let 



a = ai, a'=—a,, b= —b,, b'=—bi, 



then u' — u = 2 («, + «2) sin" - + {bi — b^) sin a. 



Jit 



The quantities «, a., 5, hi are to be calculated from the values found 

 for p and Q corresponding to any particular value of a, which I have 

 taken, for example, as the angular distance from the red to the piuple ; 



.-. a = lV 46', we have also Z) = Z),„ - a = 40°. 16'. 



