160 ON THE APPROXIMATE SOLUTION OF CERTAIN PROBLEMS, ETC. [3*74
Since x and y are known as functions of £ when rj =. 0, these equations determine the A's and the JB's, and the general values of x and y follow. When f = 0, but rj undergoes an increment,
-..., ............(14)
* + ..., ............(15)
in -which we may suppose 17 = 1.
The A's and J3's are readily determined if we know the values of an arid y for 77 = 0 and for equidistant values of £, say £ = 0, £= ± 1, £= ± 2. Thus, if the values of oc be called #03 a?_1} a?la «2> *-s, we find
J.0 = £c0j and
At \ ( \ A. ^ ^—2 *^1 *^'—1
•n.\ — TT (.#1 W—l) ~ TK (^2 "~ ^—2/j -"-a = -j"o p )
3 24
I M _r_u O/y* m „[, #7 — 2'
24 6
The 5's are deduced from the J/s by merely writing y for x throughout. Thus from (14) when £ = 0, 77 = 1,
0 / — v JL / b
(16)
5 1
Similarly y = y0 - (^ + i/_j - 2yfl) + (2/2 + 3/-2
(17)'
By these formulae a point is found upon a new stream-line (17 = 1) corresponding to a given value of £. And there would be no difficulty in carrying the approximation further if desired.
As an example of the kind of problem to which these results might be applied, suppose that by observation or otherwise we know the form of the upper stream-line constituting part of the free surface when liquid falls steadily over a two-dimensional weir. Since the velocity is known at every point of the free surface, we are in a position to determine £ along this stream-line, and thus to apply the formulae so as to find interior stream-lines in succession.
Again (with interchange of £ and 77) we could find what forms are admissible for the second coating of a two-dimensional condenser, in order that the charge upon the first coating, given in size and shape, may have a given value at every point.
[Sept. 1916. As another example permanent wave-forms may be noticed.]e, so that at all stages of the appro ination the calculated value of (14) exceeds the true value of (15). In t application to a condenser, whose armatures are at potentials 0 and