338 G. H. KNIBBS. 
and j say }', we have for ( = 85°, v = 112 cot (= 102. We 
may therefore, as before indicated, and without being involved in 
an error of even 0°01, always take £ as the angle between the 
horizontal wire and the vertical through the intersection J of the 
inclined wires, and x as the angle between the inclined wires and 
the same vertical. This simplifies the solution. 
Through the point I, Fig. 3, draw In parallel to the vertical 
through J, and I m at right angles thereto: then = angle PIn,’ 
and x = angle QIn. The angles at O, I, P and Q of the quadri- 
lateral are respectively y + x +y, 180°—-y, 90° - a, and 90° -y. 
If the line I P be inclined upwards, we write — for +a in the 
first of these quantities and vice versa in the third: the others 
remain of course unchanged. By an appropriate construction” 
the values of In, Im may be written down almost by inspection. 
Calling the former, that is the vertical one X, and the latter or 
horizontal one Y, and the lines OC P, CQ respectively S, and Ss, 
the result, after some slight simplification, is 
X=S;, cos y (sin x—cos x cot y) + S, cos # cos x cosec 7 \ (38) 
Y =8S, cos y sin € cosec y+ 8, cos x (—sin £ cot y ¥ cos £) 
In the last term of the value for Y the lower, #.¢., the + sign, is 
to be taken, if the point P is above the line Im: the minus sign 
is for the case illustrated in the figure. _X is unaffected as regards 
its signs; the variation in the angle y consequent upon variation 
in the direction of I P, produces the requisite modification of its 
value. 
If the line I P be within say two degrees of the horizontal, 2€ 
lies between 176° and 184°: hence from (37) we see that « will 
not be numerically greater than ;y of 4°, that is than 6’, when 
the zenith distance is 85° or less. The error of putting 1 for its 
cosine is consequently not greater than about one half millionth. 
oo ia ee 
1 Or its supplement, the former in the figure. It is perhaps somewhat 
safer to take it always as shewn. 
2 As for example, by dropping the perpendiculars CU, CV say—not 
shewn in Fig. 3—from C on to I P and IQ, and again from U on to CV, 
U W say, and from I on to U W. 
