9 
the treatment of geometrical problems, because it requires no 
previous selection of axes, rectangular or other. The author 
is, indeed, aware that the cooperation of other and better ana- 
lysts will be necessary in order to bring the method of quater- 
nions to anything approaching to perfection. But he hopes 
that an instance or two of the facility with which some ques- 
tions at least allow themselves to be treated by this method, 
even in its present state, may not be without interest to the 
Academy. And he conceives that two examples in particular, 
one relating to the composition of translations, and the other 
to the composition of rotations in space, may usefully be se- 
lected for statement on the present occasion. 
As preliminary illustrations of the operations employed, it 
may be remarked that for any system of lines having direction 
in space, it is required by many analogies (and is, for lines in 
one plane included among the definitions or results of the 
theories of Mr. Warren and Mr. Peacock), that the sum should 
be regarded as being equal to that one line which constructs 
or represents the total effect of all the different rectilinear 
motions which are expressed by the different summands. 
' Vectors are therefore to be added to each other by a certain 
geometrical composition, exactly analogous to the composition 
of motions or of forces, and following the same known rules, 
_ Scalars, on the other hand (that is to say, the so-called real 
parts of any proposed quaternions), admitting only of a pro- 
gression in quantity, and of a change of sign, without any 
other changes of direction, are to be added among themselves 
by the known rules of algebra, for the addition of positive and 
negative numbers. ‘The addition of a scalar and a vector to 
each other can be no otherwise performed, or rather indicated, 
than by writing their symbols with the sign + interposed ; 
each being, as we have seen, in some sense, imaginary with 
respect to the other. These operations of addition are all of 
_ the commutative, and also of the associative kind; that is to 
_ say, the order of all the summands may be changed, and any 
