1897.] near the Focus of a Telescope. 201 



image, gives no indication of any well-defined ring at the outer 

 boundary. It shows also no reason for thinking that, when the 

 values of y and z are larger than for those curves, the intensity 

 curve takes a kind of limiting form at its boundary which is 

 independent of the value of y, and yet this is certainly indicated 

 by the photographs. 



In the following I have attempted to find an approximation, in 

 the general case, to the values of the series V and V x when y and 

 z are both great and their ratio nearly equal to unity. 



Expressing J n (z) as an integral by means of the formula 



(i) n J n (z) = I e iz cos * cos ii(f) d<j>, 

 Jo 



and writing — = k, where k < 1 , we have 



V +iV x = J, (z) + ifcJ x (z) + Pk 2 J 2 (z) +... 



1 f" 

 = - e fe008 * (1 + k cos (f> + k 1 cos 2(f) + . ..) d<f> 



7T Jo 



1 C v {scos( j, 1 — K COS <f> , 



7T J 1 — 2/e COS <fi + AT 



= — I e iz& >** \ 1 + . * . , [ # 



Z7T J o ( L — 2/C COS (p + K-) 



- 2 J «^+ 2 7T Jo 1-2/ccos^ + zc 2 ^ 



Now consider the integral 



J /U, — COS 



If fx > 1 it may be expanded in the form 



1 [ n ■ * /-. cos <b cos 2 <f> \ 7 . 



_ e wcos* l + _ r + — -2:+... )d4> 



/u, \ ip dz i-/M- dz- ) J o 



7T / , lfZ 1 rf" \ r / \ 



yu, V *a* »£ i-p- dz- 1 



d n J (z) 

 Since u, > 1 and — 7 ° can never exceed a certain finite 



