274 Proceedings of Royal Society of Edinhurgh. [sess. 
Coordinates. 
Taking 0 as origin, the co- 
ordinates of A, B, C are taken to 
be 
respectively. Then 
xx^ -f yy^ -f -= 0 , 
xx^-\-yy^^zz^ = ^-, 
whence 
x\y:z = y^z^ - y^z ^ : z^x^ - z^x^ : 
^1^/2 ” ^2^/1 • 
Quaternions. 
Points A, B, C are determined 
by their vectors a, /?, y. Then 
Say = 0, S/?y = 0 ; 
whence 
my = Yaj3 , 
m being an arbitrary scalar. 
Here to compare the two solutions, observe that the two equations 
Say = 0, SySy = 0 are in fact the equations xx-^-\-yy-^ + zz-^ = 0j 
xx<^-\-yy^-\-zz^ — ^\ and so also my = Ya^ denotes the relations 
x:y :z = y-^z^ - y^z-^ : z^x^ - z^x-^ : x-gj^ - xgij^. But a quaternionist says 
that my = YaP is the compendious and elegant solution of the 
problem as opposed to the artificial and clumsy one x:y :z = 
ViH ~ V2h ■ ~ • ^12^2 “ ^2Vv I join 
issue ; my = Yap is a very pretty formula, like the folded-up pocket- 
map, but, to be intelligible, I consider that it requires to be 
developed into the other form. Of course, the example is as simple 
a one as could have been selected ; and, in the case of a more com- 
plicated example, the mere abbreviation of the quaternion formula 
would be very much greater, but just for this reason there is the 
more occasion for the developed coordinate formula. To take 
another example, the condition, in order that the vectors a, /3, y may 
be coplanar, is Sa/3y = 0, and Professor Tait contrasts this with the 
prolixity of the corresponding coordinate formula 
a?, y, z 
Xij 2/ij 
^2) ^2 
= 0 
I remark that, when all the components of a determinant have to 
be expressed, nothing can be shorter than this, the ordinary deter- 
minant notation, which simply expresses the several components in 
their line-and-column relation to each other. But as a mere abbre- 
