TRANSACTIONS OF THE SECTIONS. 3 
tion at successive instants of time, and of distinguishing, at a second point in the 
course pursued, a direction identical with a given direction at a former point. He 
will accordingly be able to form a conception of the path traced out by the genera- 
ting point moving continuously in a single constant direction, or returning again to 
the same constant direction after having quitted it, and moved for awhile in a path 
of any other description, and he might thus originally acquire the notion, in the for- 
mer case, of a straight line; in the latter, of parallel straight lines. 
Again, the possibility of drawing a straight line direct to any given point in space, 
involving (as it does) the identification of points determined by different paths di- 
verging from a common starting-point, supposes in the student the capacity of esti- 
mating the distance virtually advanced in the direction of given coordinates whilst 
actually moving in a path of any description. Hence the fundamental standard of 
position expressed in the following definition. Certain points are said to coincide or 
occupy the same position when the whole space by which respectively they are separated 
- from a point antecedently known is identical in respect both of distance and direction. 
— 
_—- 
—S 
= 
ee a 
In the conception of superficial figure the inclination-at each successive point 
(measured by the direction of the resistance, supposing the surface to be that of a 
hard body) is analogous to the direction in the case of linear figure. The superficial 
figure corresponding to a straight line will accordingly be a surface opposing (when 
solid) at every point an absolute resistance to pressure in a certain constant direc- 
tion, and the fundamental definition of a plane will be a surface passing through every 
point which can be reached from a given point under the condition of a total negation 
of motion in a certain constant direction. 
Exposition of a System of Quaternions. By Sir Witt1AM R. Hamitton, 
In this system letters did not mean quantities but directions, and the operations 
analogous to addition, multiplication, &c., had neither the same meaning, nor were 
they governed by the rules of these operations in common algebra. It was to an 
explanation of their meaning and exemplifications of the application of this calculus, 
and the facility which it afforded of arriving very simply at results of difficult attain- 
ment by the ordinary methods, that Sir W. Hamilton confined himself in this com- 
munication. Sir W. Hamilton said that he wished to have placed on the records 
the following conjecture as to a future application of quaternions: is there not an 
analogy between the fundamental pair of equations, 77 =k, ji = — k, and the facts 
of opposite currents of electricity corresponding to opposite rotations? 
_ A description of a Machine for finding the Numerical Roots of Equations, 
and tracing a variety of useful Curves. By ¥. Basurortu, B.A., Fellow 
of St. John's College, Cambridge.’ 
The machine described is capable of tracing curves whose equations are of the form 
e@=acos (mb + a) +6 cos-(nd+ 8) +c cos (r§+ vy) +&c. &c., 
or e=Zacos (md+ a), 
where abc....mnr.... are integers or fractions. 
It may be applied to the solution of equations in the following manner. Suppose 
that the proposed equation is 
Pot” + Pwo... +Py=0,- 2. + + (@) 
_ where pop, . - - - pn and represent known numerical quantities. For all values of 
2 not beyond + 1 and — 1, we may make 
x = cos 6. 
Substituting in («.), we get 
pp cos” 6+ p, cos” A+ ...... Shp SSO pS hat Gey 
and expanding cos” 6, cos”—'§...... in series of cosines of multiples of 4, we get 
an equation of the form 
Ain go cos nb+ 9, Cos (M—1)O+..064- +9, =0. - « (y) 
B2 
