315 
of Edinburgh, Session 1870-71. 
constant coefficients (equivalent to three simultaneous linear 
equations in scalars of a very general form) 
P + <pp + = 0 , 
where <p and if/ may, or may not, be self-conjugate. 
If they be self-conjugate, this represents oscillation under the 
action of a force whose components, in each of three rectangular 
directions, are made up of parts proportional to (though not neces- 
sarily equimultiples of) the displacements in these directions. The 
resistance parallel to each of three other rectangular directions 
depends in a similar manner on the corresponding components of 
the velocity. 
The operator in the left hand member may be written 
f-s + ’M* 
It + x) ( 
dt 
suppose, where x and 6 are two new linear and vector functions. 
Hence, comparing, we must have 
X + 0 = <p 
xo = 
or, eliminating 0 , 
X 2 + t = x9 
a curious and apparently novel species of equation from which to 
determine the function 
[We might have arrived at it, by a somewhat more perilous but 
shorter route, by assuming as a particular integral of the given 
equation the expression 
P = •“**■] 
If we take their conjugates in addition to the two equations 
connecting 6 and y, we see at once that all four are satisfied by 
assuming these two functions to be conjugate to one another, pro- 
vided <p and kJ/ are self-conjugate. Hence in this special case we 
may write 
x = if + v - e 1 
V.e/' 
It only remains that we should find e, and the rest of the solution 
is to be effected as in (4) or (5). 
