Captain Owen on Circummeridian Altitudes. 173 
great celestial spherical triangle (P Z ©) formed by the elevated 
pole, the zenith, and the object. Supposing, in Fig. 1, the object 
_to change its place from © to 6 by a change of altitude—=ab and 
a change of time= Zt= Z O P&B, these two changes (when P Z, 
the colatitude, and P © (= Pb) = codeclination, both remain con- 
stant) must also cause a change in the azimuth= Zz= ZO Z), 
and the change in Z© zP may be esteemed also to be the change 
at ZPOZ of position, for ZPOZ=ZPbZ+Z6ZO nearly. 
It may be remarked, that if the triangle had been supposed to 
change to the middle point C, and to become PC Z, whose quan- 
tities (for our purposes in calculation where the differences are 
small) may be considered permanent, as L’, the colatitude, and D’, 
the codeclination, are really so by hypothesis, and the effect of 
the changes will be, one half on one side of C, and one half on 
the other; but in similar triangles, the halves have the similar ratios, 
or relations to each other, as the wholes, therefore we need only 
consider them in the differential triangle Oba right angled at a. 
Wherein © )=¢= interval of time in are X cosine D, the declina- 
tion, ab —change of altitude in arc of same denomination as ¢, 
© a=z=interval in azimuth, in arc (same as ¢) X cosine A, the 
altitude. Now any two of these three corresponding changes, rep- 
resented truly by the three sides of the triangle ©ab, not only 
may the other be found, but also the angles a0b—=ZPOZ of 
position and its complement ZOba. Then the following analo- 
gies are evident, viz. where © b>=¢'=the reduced interval of time 
©a=z'= reduced interval of azimuth, and a=difference of altitude, 
HSER eo wadest scisl ZO treeidecdsii LZ'O: 2 
Also removing the differential triangle to Z, nn Fig. 2, we perceive 
the same relations, except that L* is substituted for D’, and angle Z 
for. Z ©, and t?==a? 44075 alsowt se rads 22) a: 3) Sin. Zt) 240" ,s/ cos. 
