504 REPORT—1890. 
be deviated from its course, as shown by HO’. When, however, the prism 
is rotated, this line will describe a cone, or, as projected upon the distant 
view, a circle, which fig. 8 may be taken to represent, the point Y corre- 
sponding to the position of the prism shown in full lines, and the point x 
to the position shown in dotted lines. The vertical motion of the image 
of the object viewed through the refracting prism is of no consequence, 
because the instrument may be directed up or down so as to observe upon 
any level. It will be evident that in this way 180° of angular rotation 
of the prism in its own plane might be utilised for setting out different 
angles by the instrument ; but it is better to restrict the motion to some- 
thing like 120° in all—60° on each side of the mean position of the prism 
when its thin edge is horizontal; this is shown in fig. 8, the motion 
of the prism being restricted by suitable stops, so that it cannot pass 
-beyond the division 6 on the one side of the upper scale and the division 
20 on the other side. This restriction of the motion possesses the advan- 
Fig. 7. Fig. 8. 
tage that the greater part of the horizontal shift produced by the 
prism is utilised, while the vertical motion is very little. 
To understand the graduations of the scale it will be simplest to sup- 
pose that the marks on the two instruments and the distant object are in 
the same horizontal plane. If, then, the thin edge of the prism is 
horizontal, the angle set out by the variable-angle instrument will be 
simply that due to the reflecting prism, fig. 7; whereas, if the refracting 
prism be rotated from the previous position through an angle 6, the angle 
set out by the instrument will be different by an amount represented 
by = 6 sin 6, according to the direction of rotation, where 8 represents 
the angle of deviation of the refracting prism. Let, then, a be the angle 
set out by the fixed instrument, and B that set out by the other instru- 
ment before the prism is introduced. Then for a position 6 of the re- 
fracting prism (defined as above, and reckoned positive when it increases 
the angle set out by the instrument) the sum of the angles set cut by 
the instruments will be A+3B+6 sin 6, and the supplement of this angle 
will be the angle subtended by the base of observation (of length b) at 
the distant object. The range will therefore be, for this position of the 
