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XXXII. On the Differential Coefficients and Determinants of Lines, and their Application 
to Analytical Mechanics. By A. Cohen', Lsg^. Communicated hy Professor Stokes, 
Sec. B.S. 
Eeceived May 8, — Bead June 19, 1862. 
Chaptek I- 
1. I PKOPOSE in these pages to prove the principal theorems of dynamics in a manner 
■which appears to me both simpler and more methodical than that in which they are 
generally proved; and I believe that I shall be able, by applying a few conceptions 
which spring naturally from the principles of higher algebra and statics,Ao give a clear 
interpretation to most of the more complicated formulae in dynamics, as well as to the 
several analytical steps which lead to those formulae. 
2. There are many reasons why the diagonal AD, which is constructed on the straight 
lines AB, AC, should be considered as the sum of those two lines. Those reasons may 
be found developed in De Morgan’s ‘ Double Algebra,’ in Warren ‘ On Imaginary 
Quantities,’ and in the Tract of Benjamin Gompertz ‘ On Imaginary Quantities.’ 
I shall therefore call AD (AD being the diagonal of the parallelogram constructed on 
AB and AC) the complete sum of AB and AC, and the two lines AB and AC will be 
called the components of AD. Moreover, denoting AB, AC, AD by P, Q, E, respectively, 
I shall express their relation to one another by the equation 
Il=(P)+(Q). 
I shall likewise denote by (— Q) a line equal and opposite to Q, and define (P)— (Q) 
to be the same as (P)4-( — Q), calling it the complete difference of P and Q. 
All lines which have the same length and direction will be considered as equal to one 
another, so that any line is equivalent to a line through the origin having the same 
length and direction. 
3. It evidently follows from the above definitions, that the complete sum of AB and 
BC is AC, and the complete difference of AB and AC is BC. 
4. Suppose now Q to represent a line which varies with the time t both in length and 
direction. The complete difference of the two consecutive values of Q after an increment 
of time Lt may be called the complete increment of Q, and may be denoted by A(Q). 
Moreover, if we divide the length of A(Q) by bd, and take the limit of that ratio, then 
the line which has that limit for its length, and which has for its direction the direction 
of A(Q), when A^ diminishes without limit, will be called the complete difierential 
coefficient of Q, and will be denoted by D^(Q). 
5. It will be sometimes found convenient to denote a line of length r, which is parallel 
or perpendicular to a line P, by (r || to P) or {rM to P). Moreover, if n represent a 
MDCCCLXII. 3 s 
