476 
THE REV. B. BRONWIN ON THE SOLUTION OF 
Let us next take the equation 
( 22 .) 
the arbitrary functions being put in the second term, for this only amounts to 
changing them and X. 
Assume 
(?!r„+(2w+l)^) V. 
Proceeding exactly as heretofore, we find 
if 
Making 
V VTa^-k ) 
by («.) the last becomes 
v-\-pfv=S.^. 
In the next two examples we shall put the equations under a somewhat different 
form, on account of their complexity. Suppose 
'^a{'^a + kX'ZTa + (3n + 2)A:) 
— A)/(CT’ — 2^)TO-('Cr + A)('5J + 2 k) 
Here we assume 
the common difference of the factors being Zk. 
If 
_pJ(3w + 2)A:\~*^ f '^a + 2k 
X. = P 
J 
_ / 'UTa + 2k \ j . 
M e shall have in this case 
v-\-pfv—X„ 
which may be replaced by three equations, each containing the first power only of f. 
As a last example let 
I /(ct)/(ct— ^)/(ct-2^)ct(ct-4-^)(ct + ^ k ) 
‘U!a['^a+k){zSa + i^in+2)k) 
Here 
u 
and also 
pT^’‘u=^. 
v-\-pfv=^^. 
This series of equations may be continued at pleasure ; and it is obvious that in 
the whole series, if we change zs into tt', and r into r'"*, the resulting equations may 
be reduced by {h.) exactly as these have been by (a.), and that the solutions or 
reductions of the one series may be obtained from those of the other by the conver- 
sion of symbols. 
