530 Dr. J. W. Nicholson on the Bending of 



This value of x tends to unity as sin 2 and 1 — 2ccos<£ + c 2 

 tend to equality, so that c = cos <£. The value of </> n used above 

 then ceases to be valid. 



Now it is clear from the form of the integral in the 

 operation g(6) that the important part of the range of inte- 

 gration is near the upper limit cf> = 0. The important 

 harmonics are therefore clustered about the value of n 

 given by 



x = sin 6/(1 — 2c cos d + c 2 )K . . . (42) 

 This is unity when 



c = cos 0, or rcosd = a. 



The plane r cos 6 = a is therefore inside a region of space 

 in which the important harmonics have an order nearly equal 

 to z. It has already appeared that v' cannot be zero for any 

 of the four component series of yp within such a region. 

 The real effect of this value of x is to make the equal 

 functions V\ and vl very small, although not zero. It 

 represents the point at which they attain their minimum 

 values, and will be considered later. This region will 

 henceforth be called the " region of transition," between, 

 as will afterwards appear, brightness and very complete 

 shadow. 



When 6 is not very small, and x is defined by (20), the 

 value of n corresponding is of the same order of magnitude 

 as z. In this case, the important harmonics are of very high 

 order, and the zonal harmonic may be replaced by its 

 asymptotic expansion. Writing therefore 



so that the leading term of dF n /dfjb is 



~" ( — =-x7i) sm (™0 — Jw), 

 \7rsm 3 0/ v 4 n 



it is found that 



yp = S" uie^-^ + e^-^+^sm (m0-±ir) 



where 



_ im (2m sin #R n Il n A 2 

 — ka 2 \ 7r / 



Again rejecting two series which cannot have a vanishing 



