AND DETEEMINANTS OE LINES. 
507 
Hence 
da , 
a formula which is generally deduced as the result of somewhat long analytical work. 
72. Secondly, in order to give a striking example of the manner in which the theory 
of the determinants of lines explains and shortens analytical processes, I will give the 
following direct proof of Euler’s equations. 
Take the principal axes at the fixed point as the axes of coordinates. Let v^, be 
the components of the velocity V, andj^j,, fy^f^ those of the acceleration F of a particle 
[x, y, z) of mass m. 
We have seen that the fact of the acceleration being the complete differential coeffi- 
cient of the velocity leads at once to the three following equations : — 
n dvx . ^ 
n dVy . 
fy — “i” 
n dVx . 
Jz ‘V'^y'^x '^x’^y' ^ 
( 1 -) 
Putting then for brevity’s sake fl for v^THTy—VyVT^^j-'y for for 
we have 
M dVx_^f> 
1=^— dt'^J 
Jy— TV y^ 
/■— ^4-f' 
dt 
\ 
( 2 .) 
Now the sum of the moments of the moving forces about Ox equals 
.... (3.) 
Let us first investigate the expression tm ^ . Evidently 
Vx^yoT^—XrSy, and Vy=xvy^—Z7!y^. 
And as the axes m'eve with the body, x, y, z do not vary with the time, and we there- 
fore obtain 
dt ^ dt It^ ~di~^ ~dt If 
Therefore 
