315 
of Edinburgh, Session 1870 - 71 . 
constant coefficients (equivalent to three simultaneous linear 
equations in scalars of a very general form) 
p + ?P + = 0, 
where <p and i p may, or may not, he 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 he written 
suppose, where ^ and 6 are two new linear and vector functions. 
Hence, comparing, w r e must have 
X + 6 = (p 
xo = 
or, eliminating 0, 
x 2 + ^ = X? 
a curious and apparently novel species of equation from which to 
determine the function y. 
[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 = g_0c Po*] 
If we take their conjugates in addition to the two equations 
connecting 0 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 if/ are self-conjugate. Hence in this special case we 
may write 
x = + v.e] 
8 = |p - V. e J ' 
It only remains that we should find e, and the rest of the solution 
is to he effected as in (4) or (5). 
