Scientific Papers - Vi

374.
ON THE APPROXIMATE SOLUTION OF CERTAIN PROBLEMS RELATING TO THE POTENTIAL.—II.
[Philosophical Magazine, Vol. xxvi. pp. 195—199, 1913.]
THE present paper may be regarded as supplementary to one with the same title published a long while ago*. In two dimensions, if <£, ty be potential and stream-functions, and if (e.g.) ty be zero along the line y = 0, we may take
•(1)
U.
_ vf __ . _ f"
JJ 1.2.3-7
fiv _
7
•(2)
1.2.3.4.5'
/ being a function of cs so far arbitrary. These values satisfy the general conditions for the potential and stream-functions, and when y = 0 make
Equation (2) may be regarded as determining the lines of flow (any one of which may be supposed to be the boundary) in terms of /. Conversely, if y be supposed known as a function of x and -\/r be constant (say unity), we may find / by successive approximation. Thus
/«! + £.*
J y ^ 6 doc* \y
±k 7........(3)
.
36 d& dtf \y} 120 d& \y;
We may use these equations to investigate the stream-lines for which ^r has a value intermediate between 0 and 1. If 77 denote the corresponding value of y, we have to eliminate / between
1 _ iif— -L- f" _L _£ -f
A ~" yj ft j ~ -\ OA j
and whence
— -Proc. Lond. Math. Soc. Yol. vn. p. 75 (1876); Scientific Papers, Vol. i. p. 272.at j> IH coiiHtiuifc being already a large departure from the case of nature, we may perhapn as well suppose /A' — 0, and then no impressed bodily forces are called for either at rest or in motion.