5 
Now every value of 0 satisfies (14). Hence 
<p(a - b) = <p(a) - (p(Jj) = Z .... (17) 
Consequently, combining (15), (1G) and (17), we have 
cj>(a) - </>(&) - x (a) + if/(b) 
'dx l dx 
= (Z, - A')t! - (Z, 
+ (Z 2 - A 2 )a - (Z 2 - B 2 )6 
= Z + A + B (18) 
But the linear differential equation (18) can be integrated 
in the form 
a + Xj6 = X 2 (19) 
where X x and X 2 are functions of x. By means of (19), 
a and b may respectively be eliminated from (15) and (16), 
giving results which may be represented by (20) and (21) 
respectively. Then (20) and (16) will have a common 
integral which will give an available value of b, and a like 
value of a can be deduced by combining (21) and (15). 
And, a and b being so determined, we have next to substi- 
tute the resulting a — b for 0 in (14) if any constant remains 
arbitrary. If not, the required integral is obtained as soon 
as a and b are known. 
I would add that, in certain cases, some of our expressions 
may become infinite at the limits, but this circumstance 
will not necessarily render the results illusory or inappli- 
cable. In dealing with the Boolian integral for quadratics 
by the method of conversion I ascertained, and communi- 
cated the calculations and results to Mr. Harley some time 
ago, that infinite values occur in the conversion but do not 
affect the final results. 
Brisbane, Queensland, Australia, 
August 5th, 1870. 
Mr. Boyd Dawkins, F.RS., gave a short account of the 
work done in the Victoria Cave, near Settle, since the last 
notice brought before the Society. The two layers contain- 
