DIFFERENTIAL EQUATIONS OF DYNAMICS, ETC. 
323 
X 
y 
;s 
X 
A2 
y 
H'o 
^2 
z 
*'2 
where Aq, &c. are functions of t, which may be either given explicitly, or implicitly 
through the variables (see the last article). 
The modulus of transformation P is found (art. 69.) by substituting for the vari- 
ables X, y, z in the expression xu+yv+zw, their values in terms of x,y,%\ we have 
therefore 
F==('koX-\-X,y+'K^z)u+{i/joX+l/j,y+f/j.,z)v+{voX+v^y-\-v^z)w; 
and then the three equations 
dV 
dy-^’ 
dP 
j-=w, give 
M =AoU-TjU/oV-j-{'QW1 rU — "hyV -\- 
=?tjU-l-j«'iV-l-j'iW L whence \ \ (85.) 
W = X.^U+[/j2V-\-V2WJ 
lw= VoM-p J'lf + 
Also we have (see Theorem VIII. art. 64.), since P is to be considered to contain t 
explicitly through Xq, &c., only, 
dP 
= (A>-f A',?/ -I-A;^) u -h (^>+^3/ +^^2-) V+ («'>+»'', 3/+ 42:) w, 
in which expression the values of u, v, w in terms of the new variables are to be sub- 
stituted from (85.). Now if we put for the angular velocities of the moving 
I 
system of axes about the axes of x, y, z, respectively, so that 
COq A2Aj “h ^ 2/^1 “h (AiA2-l~jU'i^2“l~*^l**2)5 
it will be immediately seen that the usual relations between the nine direction cosines 
enable us to put the result of the substitution in the following form : 
(^) ■\-u,{zu—xw) -\-a)^{xv—yu). 
The original differential equations 
dn 
I dZ o 
U — — -^5 &c 
dx 
are then transformed into 
f d^^ I d^^ g / 4. i:! /I \ 
x -=—, &c. (art. 64.), 
(XUi (XOL 
where 
‘»=(Z)-(f)- 
2 X 
MDCCCLV. 
