336 G. H. KNIBBS. 
and the corresponding expression in x, which may also be written 
y=XxX-X = — psin 2x (1+p cos 2 x + ete.) (x) 
Tt is convenient to replace » by terms in c¢/S, thus from (s) and 
(w) we have, on rejecting as certainly negligible the powers of this 
fraction — than the second, 
x = [sin 2 2é[1 - < (cos 2€-4)] (36) 
and a sfehar’ expression i y containing y instead of &. We 
shall shew that even the secondary terms in these equations for « 
and y, are also always negligible in the application with which we 
are dealing. Evidently the value of (36) is a maximum for € = 
45°, when cos 2 € is zero, and the whole value of the secondary 
term is +c?/2S8?, as the sine factor is then unity. An altitude 
of 5° may be regarded as the lowest at which an observation 
should be made, in which case ¢/S is about 22°°69/960" or zis: 
ome aap the error of omitting the secondary term can never 
amount to 534-5 for any system of wires, and is about that amount, 
or 0°’26 for the angle QCN in Fig. 3, a quantity which is quite 
negligible since its effect will never be more than 0-’001 in the 
final results. We may therefore always write the preceding 
equations in the simpler form 
w= %sin2é,y = % sin 2x (37) 
If x and y are required in degrees, minutes or seconds of are, the 
quantities must be multiplied by the number of degrees, minutes 
or seconds ina unit of circular measure, as for example, by 3437°7 
for the result in minutes. 
We have supposed C M to be vertical because the elliptical 
outline is then symmetrically situated with respect thereto. The 
relations of the lines I P, I Q to the verticals drawn through I or 
J—since these last are nearly identical—are ascertained by 
observation: but the convergency of the verticals, through say C 
and I, is a function of both the semidiameter and contraction, as 
well as of the directions of the lines, IP, 1Q. This convergency 
however, is considerable only for high altitudes—i.e., when the 
ellipticity is extremely sma!l—and is very small when the ellipticity 
