194 REV. M. O’BRIEN ON SYMBOLIC FORMS DERIVED FROM THE CONCEPTION 
Su 
which, putting for ^ the above value, becomes 
^ + 2D^ . +D<y . (D<y . m) . 
Hence if U denote (symbolically) the resultant of the accelerating forces, whatever 
they may be, which act on the point (n), we find 
|?+2D».§+(D‘")‘«=U (1.) 
I may observe, in passing, that, since c?+(D(y.) represents the complete differentiation 
of u, we might have written down the equation of motion immediately, in the form 
{d+(jy..)Y 
which, expanded, is identical with (1.). 
Now, if we had forgotten the earth’s rotation, we should have put, instead of (1.), 
d'^u 
dF~^- 
Hence it appears that, if we neglect the rotation in forming the equation of motion, 
we may correct the error, by supposing that there is the imaginary force 
-|2D<».|+(D<»>| (2.) 
acting in addition to the real force represented by U ; for on this supposition we find 
u, (3.) 
which is equivalent to (1.). 
As regards terrestrial problems, however, the expression (2.) admits of an important 
simplification; for the accelerating force of gravity (g), which of course is included 
in U, is supposed to be the resultant of the earth’s attraction, and the common centri- 
fugal force. Now this common centrifugal force is that which is conceived to be in 
action upon a particle rigidly connected with the revolving earth ; in other words, it 
SjUl 
is what (2.) becomes when Wherefore the expression for the common centri- 
fugal force is 
(4.) 
As this therefore is included among the forces which U represents, it ought to be 
omitted in the equation (3.). Thus we find that 
-2D-I (5.) 
is the force which must he supposed to act on the point (u) as a correctioji for the 
neglected rotation. 
We may, therefore, in all problems of motion relative to the earth, forget altoge- 
