382 
in the arbitrary functions ¢,, ¢., the expressions y+jx, z+ kr 
for y and z respectively, developing the results in ascending 
powers of x by Taylor’s theorem, and then substituting, ac- 
cording to a certain directive canon, ‘mean products’ of j and 
k for ordinary products. I think it a remarkable circumstance 
that it should be possible thus to obtain the true developed 
values of F\, #,, F;, and the theorem upon which the process 
is founded is well worthy of being recorded. But I cannot 
agree with you that it can be considered as virtually freeing 
your solution, viewed with reference to the determination of 
F,, F., F,, fromimaginary quantities. For in any parallel case in 
which imaginary quantities are involved in an algebraic expres- 
sion, we can, formally at least, get rid of them by substituting 
for the given expression some other expression not involving 
those quantities, with the provision that after development 
certain changes, governed by a particular rule or canon, shall 
be made. ‘The function cos, for example, considered ana- 
lytically, involves imaginary quantities, for it is expressed in 
v¥=-1 de e-r-1 
finite terms by the formula - —. Now I apprehend 
that we.should not virtually escape from this condition of its 
finite expression, by presenting the function under the form 
x ~x 
2 > , and adding, as a direction, that in the development 
of this function the signs of the alternate terms should be 
changed. 
‘* When among the physical conditions of a problem de- 
pendent upon the differential equation 
du a du ae | 0 
dx? dy? dz ° 
symmetry, with reference to an axis, as that of x, isinvolved, — 
a particular integral of the equation may be presented in the 
form wu = [' d0p (a+r cosO0y —1), wherein r= (7 + 2*). 
This integral was given first, so far as I am aware, by myself 
