162 REV. M. O’BRIEN ON SYMBOLIC FORMS DERIVED FROM THE CONCEPTION 
Fig. 1. 
• C 
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Fig. 2 . 
of the kind called ^'^distributive" and from its distributiveness all its properties in a 
great measure follow as necessary consequences. Now this is a fact of great import- 
ance to be borne in mind, a thing to which the truly mathematical adage when 
found make a note of" fully applies. There is no notation actually in use for this 
purpose ; but there are two nearly obsolete signs, which by an easy and natural 
generalization, and according to the most approved rules of mathematical interpreta- 
tion, might be made to serve effectually as representativ^es of the effects produced 
generally by the translation of a directed magnitude. My design, in what follows, 
is to show this, and establish the laws according to which these signs are to be used 
in their enlarged signification. Afterwards I shall endeavour to justify the proposed 
innovation, if such it is to be considered, by showing its utility in a variety of cases. 
(2.) Instances which suggest the proposed symbolization . — There are three ele- 
mentary conceptions which have suggested to my mind the principles which it is the 
object of this paper to develope, and they will serve here as means of introducing 
the subject, and furnish the best foundation to build upon. They are the following : — 
] St. The generation of surface by the parallel motion of a right line, 
of which the simplest instance is a parallelogram ABCD supposed 
to be generated by the motion of AB, parallel to itself, along AD. — 
2ndly. The effect produced on a rigid body hy the translation of a force ^ 
acting upon if from one point A to another B, the direction of the 
force remaining unaltered ; which effect, as is well known, consists 
In that peculiar tendency to motion that results from the action of 
the couple composed of the force at B, and a force equal and oppo- 
site to the original force at A. — Srdly. The effect produced hy the 
translation of a force resulting from the actual motion of its point 
of application ; which effect is now usually designated by the term work 
point of application be supposed to describe the path ADB, the force 
all the time acting parallel to its original direction, a certain amount 
of work is accumulated in consequence of the translation of the 
force, and this is the effect I allude to. 
Now in each of these cases the conception in the mind is that of 
the effect produced by the translation of a directed magnitude ; and 
what is worthy of special remark is this, that in each of these cases the effect alluded 
to is represented hy ihe product of two factors, one h^mg the translated magnitude, and 
the other the amount of translation it undergoes. Thus, in the case shown by figure 1, 
the surface generated is denoted by the product of AB into the perpendicular or 
lateral distance between AB and CD ; of which, AB is the translated magnitude, 
and the perpendicular the amount of translation that takes place laterally. In the 
case shown by figure 2, the effect is also represented by the product of the translated 
magnitude, that is, the force, into the amount of lateral translation. In the case 
shown by figure 3, if we suppose BC to be the direction of the force produced back- 
If the 
Fig 3. 
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