450 
PEOFESSOR BOOLE ON SIMTJLTANEOES DIEEEEENTIAL 
Applications. 
1st. Suppose it required to find the number of integrals of the form P=C 5 which the 
system of difierential equations 
dz—{t ■\-xy-\-^z)dx-\-{3czt-\-y—xy)dy, 
dt—{y-\-z —‘^x)dx-^{zt —y)dy 
admits, and to determine such integrals. 
Eliminating dz and dt from the equation 
-d^dx+^dy+d^ dz-\- dt dt—^, 
and equating to 0 the coefficients of dz and dt in the result, we have 
( 1 -) 
(2.) 
Hence writing 
^ =S+(''+-'^+“)s+(3'+^-3*)5. 
dz 
dV 
and forming the equation 
(AA'— A'A)P=0, 
we have, on rejecting a common algebraic factor, 
dV dV „ 
**+w=o- 
(3.) 
did 
By substituting in (1.) and (2.) the value of -yj hence obtained, we have the system of 
three equations, 
S+(3x'+i)S=0, 
dV ^ dP „ 
dP. 
dt^^ 
dz 
d^ 
dz 
= 0 . 
Now if upon any two of these we repeat the same process as upon (1.) and (2.), we 
obtain as the result 0=0. Thus the system of partial difierential equations is complete. 
As then there are three equations in this final system, while the number of original 
variables was four, the primary system will admit of one integral of the form P=c. 
^p 
To obtain this integral, eliminate -yy from the above equations and 
dP ^ , dP , , dP^^ ^ dP 
