Wave Resistance: Effect of Parallel Middle Body. 81 
Using the notation 
wols 
D(a) = (=ilp | cos” ¢ sin (p sec $) de, 
wv 
Ban) = (ee [coe $ cos (psec ¢) dd, (9) 
we have 
__ 64geb7l &y 118 & (gl (2) 
Bis Te gl E Tis gl oP \e a) = pp ts Ce 
* e Ae @ 
4P3 (pi) — gi Px (pi) 12 op Ps (p1)-++ gi Pa (pz) 
4 
4 
— agp Pole) +4 Ps ks) | (10) 
where 
pPi=g(2k+2le, p2=g(2k+D/c*, ps = 2gk/e°. 
Tabulation of Functions. 
5. In order to obtain the curves we require, we have to evaluate (10) for a 
large number of values of &, and in each case for several values of ¢; it was 
necessary to prepare tables and graphs of the P functions. Yo and Y, being 
Bessel functions of the second kind and Yo + being defined by 
Meu = ['¥%o (p) dp, (11) 
we have, from sequence relations given previously,* 
== a (p?—3p) Yo *+-p"Yo—(p*—4p) Yi} 
Py=— 5 {(P*— 6p? +9) Yo "+ (p*—9p) Yo—(p*— Tp") Va} 
Pp ai {(p°—10p3+-45p) Yo *+(p'—13p") Yo—(p° — 11 p+ 64p) Yi}. (12) 
The values of Yp and Y, and of Struve’s Functions Hp and Hj, given in G. N. 
Watson’s ‘‘ Treatise on Bessel Functions,’ were used to calculate values of 
Y, 2 from the formula 
Yo + = pYo— + p(YoH:— Yio), (13) 
* «Roy. Soc. Proc.,’ A, vol. 103, p. 578 (1923). 
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