344 G. H. KNIBBS. 
Fig. 3 in§ 14. With telescopes of high power the types of obser- 
vation marked 1 and 2 are alone possible, since the field of view 
is very limited: with low powers, type 3 is also possible. Some 
practice is necessary to observe simultaneously the two tangencies 
of this last type ; the observation of a single tangency is much less 
fatiguing. It has already been pointed out however in § 16, that 
one of the results of the single tangency observation is not very 
reliable ; for example, the positions marked 1 in Fig. 4 determine 
the altitude well, and the direction indifferently: 2 on the contrary 
gives good results for direction, and indifferent ones for altitude. 
With an electrochronograph the difficulty arising from this cir- 
cumstance can be obviated as hereinafter shewn, but without such 
an accessory it may he necessary to determine which of these types 
of observations, 7.¢. 1 or 2, is to be preferred. This preference 
will depend really upon three things, viz.:—(i) the ratio of the 
probable error of the measurement of a zenith distance to that of 
an azimuthal direction; (ii) the ratio of the variation in zenith 
distance to that in azimuth, and (iii) the ratio of the probable 
errors of the estimation of horizontal and vertical tangencies. 
We may omit the consideration of the last, and suppose that an 
investigation has shewn the first (i) to be x, a quantity which is 
usually greater than unity and in some instruments amounts fully. 
to 2. Then, for an observation for meridian, we should prefer 
1 if = < x, and.2 if as > Ke 
d¢and dA denoting respectively the relative motions in zenith 
distance and azimuth. The reason of this is evident. In the 
former case it is necessary to measure the altitude with the greatest 
possible precision because of the large influence an error therein 
would have on the result. In the latter the error is less serious; 
and it is advantageous to bestow more attention upon the measure 
ment of the azimuth. As before indicated, it is assumed that the 
defect in the estimation of the tangency is about the same for 
both instances, which is probably true in the case of observers 
