602 Messrs. Aichi and Tanakadate : Theory of 
2\3 2\3 : a 
eae oe ae. _K (4% 
22y (5): ey (ZY K K’(<). 
and y’ = —y in the latter half, 
Putting 
then applying the well-known sequence equation of the | 
Bessel’s functions | 
ute) =3{J2(w) +Jo(w) 
this can be reduced to the form 
i} { du'dy’ cos {x + y°—K’O(a! +y') } [Jo{ Kw’ + y/)} + To{ K(x’ +y)}], | 
0 —0O 
which is almost intractable for practical calculation. 
4. It will be advantageous to consider first the maxima and 
minima of F as compared with those of f2, and then to discuss 
the general character, and finally proceed to the numerical 
calculation of F. For this purpose, transform the variables 
$, ¥ to wv, y, which are given by e=¢ cos, y= sinw, and 
then integrate with respect to y. Putting Kz=z2, we arrive 
at the expression 
+Ko 
F(K, ©, K@)= m dz,/(Koy—2f(K0 +2). 
CP 
ee ( 
K@b?k? 
cc — 
The maxima and minima of F are given by 
a) i 
4 _ ew (KO)? = 2 56 {f°(KO + 2) } =0; 
de @. 5 hy nD Oe SR 
i dzn/ (Kp)? —22 5 { f?(KO+2) }=0. 
—Ko i 
Or, putting the mean value of / (Kq)?— 22 in the integral, 
we find the approximate relation 
r*(KO—K®) =/*(K0+ K®), 
For the smaller values of 6, especially at the first maximum, 
f* has no symmetry on both sides of the maxima and minima : 
so that the first maximum of F receives small displacement 
towards @=0 as compared with /”. This displacement becomes 
smaller and smaller for other maxima and minima. For 
larger values of @, as ¢? is nearly symmetrical on both sides, 
the maxima and minima approximately coincide with those 
of f?; nevertheless it does not follow that the maxima for f 
always remain as maxima for F. 
