735 
of Eclinbitrgh, Session 1883 - 84 . 
instantaneous configurations of the points A , B , and C , and the 
situations of the fixed lines of motion AA and BB relative to those 
future configurations. So at any future instant when one of those 
prophesied configurations arrives, we shall have the means of speci- 
fying relative thereto the lines AA and BB ; and these, as already 
explained, are to be accepted as the desired frame ; that is, as being 
a frame accomplishing all the requisites of the problem. 
Method II. — Ee verting now to the stage arrived at just before 
our commencement of branching out into the two varied methods, 
and maintaining the state of things there attained to, we may pro- 
ceed by a second alternative method as follows : — It is to be recol- 
lected that, in the state of things attained to, we have three templet 
points Aj , Ao , Ag , all in one straight line AA ; and three other 
templet points B^ , B2 , Bg , all situated in another straight line BB ; 
and that, while keeping the line AA at a changeless distance from 
the point C, and regarding the line AA and the point C as being at 
rest, and consequently regarding the points Aj , A2 , Ag , as being also 
all at rest, we can keep moving the line BB, in the one freedom it 
possesses, and so we can alter continuously the distances Bj^B2 and 
B^Bg , and we may readily further see that we can alter continu- 
ously the ratio of either of these distances to the other. So let us 
ordain the requirement that we are thus to keep moving the line 
BB in the freedom it possesses till we attain the condition that : — 
As AjA 2 : B^B2 : : A^Ag : B^Bg . 
For the purpose of accomplishing this requirement we may use 
(ideally at least) a mechanism of parallel rulers or parallel pro- 
jectors, &c., which will now be briefly described. For guidance in 
geometrical principles towards formation of the conceptions intended, 
imagine a straight line placed intersecting the lines AA and BB. 
Let us name this line as the transversal, or refer to it as the line 
XT. Imagine the points Aj, A2, Ag, to be projected to the trans- 
versal by parallel projectors ; and imagine the points so found on 
the transversal to be further projected to the line BB. Call the 
points so found on BB, the points 62 , 6g , in correspondence 
with Aj , A2 , Ag , from which they have been respectively derived 
E"ow obviously we have by geometry, 
