of Edinhurgh, Session 1883-84. 
731 
promptly replied that he could solve it very briefly by quaternions. 
(See ante, page 575, footnote.) I myself soon after succeeded in de- 
vising a solution, and my object in the present paper is to ofi'er that 
solution to the Society, Prof. Tait too, I trust will submit bis 
quaternion solution this evening to the Society. 
Let us take the case of three points moving rectilinearly and 
mutuo-proportionally relatively to a frame. Three points are 
enough, but we might use more, and might so bring out varied 
solutions ; and besides it may be mentioned here that a distinction 
of importance will be found to exist between the results attainable 
for three points only, and for a greater number than three. This 
will be referred to at a later stage (near the end of the paper, pages 
740 and 741). 
Let these three points to be used be called in general A, B, and 
C, irrespectively of changes in their mutual configuration, or in 
their situations relative to any frame. Let us proceed to find a 
frame relatively to which any one of these three points, say the 
point C, shall be at rest, and the other two shall move rectilinearly 
and mutuo-proportionally. Let successive simultaneous situations 
of A and B at instants of measurements he designated as A^ and 
Bj , Ag and Bg , Ag and B^ , &c. Yfe may thus bring into considera- 
tion and into use a set of portable triangles A^CB^ , AgCBg , A3CB3 , 
and more if wanted, representing severally in forms and dimensions 
the likewise designated original triangles ; or, it may be, represent- 
ing the originals in form, while constructed on any convenient scale 
of dimensions. The use of such altered scale is to be understood 
as available if the full original sizes would be inconvenient for use 
in a kineniatical diagram, or mechanism, soon to be explained for 
construction or ideal contemplation, After this mere mention of 
allowable change of scale, the explanations will generally be given, 
for brevity and simplicity, as if the lengths of lines in the diagram, 
model, or mechanism, were identical with those of the corresponding 
original lines, rather than on an altered scale. We may denomi- 
nate these several portable triangls in succession as templet 1, 
templet 2, &c. 
Let us place the corners C of the three templets together, and 
take any plane passing through C, and bring the three lines CAj , 
“'CAg, CA3 , into that plane. Then take a straight line to be called 
