Mr. Gompertz's analysis applicable to the 
24 6 
^x:p,q ' L r +x 
At LT It Lt i 
- " ^ ^n:r,p,q + m 6 n:r,q,p + m , *-‘n-.r,q+m,p+in\ 
3 n:p,q,r 
— L 
3 n : p, q,r + m 
+ 6" L « : P, 
6 —n.T,q,q>-Yin 3 
+ iL„: 
/>, 7 -f 7n, r -f- m 
/> + m, q , r + m 
+ 
~3~ ^n + m -.p.q, r ■> 
This form of developement would give the solution of 
some problems, to be considered in this paper, in the form 
which has already been given by mathematicians ; but there 
are cases of application, in which the form in Article 9, will 
give a much less intricate solution. 
Article 1 x . Because L„ ;f< q — ^(L f+n • L' f + „+ L +n . V p+n) 
+ ~V p+n .Uq +n , part of an expression of Article 9, is = 
^( L p+n T 1 ’” V/ T L />+«) * * 1 • L '/>+») + 
t( l , 
■^•1+Vi.L,^ ) . (L — 
2 3 p+m ' <?+« 
• 1+ — L/ , ) = 
"p + n 2 ’ 3 p + n* ' v ?+« 2 3 ?+ 
agreeably to the hypothesis being 
= — -v/ —and k = — 4 - v' — ,and therefore the fluent 
of . L r+J . between the limits of x-=.n and x=.n + m is= 
" 2 " k ( ^n + b : p, q H” ^ n+k : p, q) ^ r+n * ^n:p, q' 
Article 12. On the fluent of L*. M>rjSf&c> x L X+JC . 
Suppose n and n-\-m sufficiently near each other to admit 
the decrements to be considered with sufficient accuracy as 
proportional to the time t in which they are produced; t 
being greater than m, and x = n -j- t : and the fluent required 
using a similar notation to that hitherto used will be from 
