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XII. On Symbolic Forms derived from the Conception of the Translation of a Directed 
Magnitude. By the Rev. M. O’Brien, M.A., Late Fellow of Caius College, 
Cambridge, and Professor of Natural Philosophy and Astronomy in King's 
College, London. 
Received April 21, — Read June 19, 1851, 
PART I. 
GENERAL INVESTIGATION OF THE SYMBOLIC FORMS. 
(1.) XhERE can be no doubt, that time and ingenuity have been often wasted in 
devising systems of notation, and new methods of algebraical representation, which 
have never proved of any service in advancing the cause of science. It is not sur- 
prising, therefore, that symbolical innovations, if they have not the strongest and 
most obvious reasons to recommend them, are generally received with little favour 
by mathematicians. At the same time, it must not be forgotten, that the mind has 
wonderfully enlarged its powers of research by the symbolization of its abstract 
conceptions, and that the various additions which have been made, from time to 
time, to mathematical notation, have contributed largely to the progress of physical 
investigation ; u'itness, for instance, the applications of the negative sign, indices, 
logarithms, coordinate equations, the differential algorithm, &c. 
A new notation, or a new application of an old notation, ought, in all cases, to be 
called for by some want in science, that is, by the existence of some important and 
often occurring conception for which there is no adequate, or at least no sufficiently 
general mode of representation. It should be neither artificial nor complicated, but 
natural and simple : it should also be based on principles of established authority, 
and framed according to allowed precedents. And, lastly, it should be capable of 
something more than mere elementary applications, and be recommended by its 
utility in the higher and more abstruse branches of science. 
With these cautions before me, and on these grounds, I venture, in the present 
paper, to propose a new use of an old notation, which appears to me to supply a 
want of considerable importance, as I hope to show by the remarkable simplifications 
which it introduces into many difficult investigations. There is an operation, if I 
may so call it, of constant occurrence in Geometry and Physics, which consists in 
the translation of a directed magnitude, that is, the parallel motion of a magnitude 
possessing the property of direction, such, for example, as a force, a velocity, traced 
line, or the like. This translation, as it may be easily shown, is always an operation 
MDCCCLII. Y 
