385 
of Edinburgh, Session 1870-71. 
“ in reality at any time t ; or merely mathematically, if &c., 
“ denote for brevity the preceding linear functions of the com- 
“ ponents of motion, the equations of motion are as follow:— 
dt 
¥<7 + Zp = X , = &C., &C.' 
-jj ~ + %V - filer + pp = L 
^ - Eu + Mw - &.*, + iLo- = M 
^ - Uv + ¥« - & P + = N 
• ( 1 ) 
“ Three first integrals, when 
X = 0, Y = 0, Z = 0, L = 0, M = 0, N = 0, 
“ must of course be, and obviously are, 
(2) X 2 + ¥ 2 + = const. 
“ resultant momentum constant; 
(3) m + fH¥ + m = const. 
“ resultant of moment of momentum constant; and 
(4) u 3^ + J¥ + + ^£L + pfK + cr= Q 
These equations were communicated in a letter to Professor 
Stokes, of date (probably January) 1858, and they were referred 
to by Professor Eankine, in his first paper on Stream Lines, com¬ 
municated to the Koyal Society of London,* July 1863. 
They are now communicated to the Eoyal Society of Edinburgh, 
and the following proof is added :— 
Let P be any point fixed relatively to the body, and at time t , 
let its co-ordinates relatively to axes OX,OY,OZ fixed in space, be 
* These equations will be very conveniently called the Eulerian equations 
of the motion. They correspond precisely to Euler’s equations for the 
rotation of a rigid body, and include them as a particular case. As Euler 
seems to have been the first to give equations of motion in terms of co¬ 
ordinate components of velocity and force referred to lines fixed relatively 
to the moving body, it will be not only convenient, but just, to designate 
as “Eulerian equations” any equations of motion in which the lines of re¬ 
ference, whether for position, or velocity, or moment of momentum, or force, 
or couple, move with the body, or the bodies whose motion is the subject. 
