576 T. H. Havelock. 
If we use only the first term in this expansion in powers of 6, we have 
J = —(20r?/ma)? be ™eF9 J 3/0 (ma). (12) 
With this value (7) gives 
DN S7z eee in mdm 
2 pp -P , ———— 
R ga be {Jape (ma) }? (mPct / g—l)? 
= 87° gpa? (1—e?)? [see her ProFsee?? {Jao (Koa sec dh) }? dd. (13) 
With ¢ nearly equal to unity, it is easily verified that (13) agrees with (8). 
On the one hand, the result (8) includes the sphere (ec = 0), under the 
restriction that fis large; on the other hand (7) and (11) give a formal 
solution for any depth f, but only serve for e nearly unity. The two methods 
are very different, and it is of interest that the results agree under conditions 
in which the two approximations overlap. 
Formule for General Type of Model. 
4. The limitations of Michell’s formula do not admit of its application to 
actual ship forms; for although the sides of a ship may be at small angles to 
the median vertical plane, the bottom of the ship does not fulfil this condition. 
It is proposed to use the method here in such conditions that this objection 
does not hold, by supposing the ship to be of infinite draught. In other words, 
we consider the wave resistance of a post extending vertically downwards 
through the water from the surface, its section by a horizontal plane being 
the same at all depths and having its breadth small compared with its length. 
This enables us to elucidate certain points of interest in ship resistance. 
We suppose the ship to be symmetrical fore and aft, and we take the 
origin at the mid-ship section. Then since in (7), /’(a, 2) is independent of z, 
we have 
ay ga co 
Es 7 le m? (mct/g?—1)” (14) 
where J= (a (x) sin ma da, (15) 
the integration covering the length of the ship, and the equation to the half- 
section being y = /(#). 
We wish to study the effect of altering the form of the section while 
keeping the length and the total displacement unaltered, the beam varying 
slightly according to the curvature of the lines. These conditions can be 
satisfied by taking the form of the water-plane section, for y positive and v_ 
ranging between +/, to be 
b a? ces 
i EN 1 
J eae =) | 6a? ec) 
204 
