382 
in the arbitrary functions ¢,, ¢2, the expressions y+jx, 2+ hx 
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 1 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-1l 4 @- 2-1 
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 
rv _ , 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 o a 0 
da? dy dz ” 
symmetry, with reference to an axis, as that of x, is involved, 
a particular integral of the equation may be presented in the 
form u = {; d0p (a+r cosy -1), wherein r= y(y? + 2’). 
This integral was given first, so far as I am aware, by myself 
