ME. A. CATLEY ON THE DOUBLE TANOENTS OE A PLANE CUEVE. 209 
where the first three columns show the numbers which give, by the addition of the 
numbers in the same horizontal line, the numerical coefiicients of the factorials which 
multiply the different terms of H, DH, &c., and where in the last column 12(1), &c. 
are written for shortness in the place of {1, 2, (1)}, &c. 
27. It is clear that we have in general 
A^D’-H= l[?^-2]'>[^^-2]’• {1, 2, (r+1)} 
+ I 2]•[7^~2]’-'{l, 3, (r) } 
4, (r- 1)} 
+ [%—!]■ 
+ 1 [?^-2]t^^-2]“ {l,r+2,(l)} 
'+R; \n—2J [w— 2]’-" {2, 3, {r—l)] 
+R; [?^-2J [w-2]’-^ {2, 4, (r-2)} 
.+K- 2 [^^— 2]'-’[w— 2]“ {2, r+1, (1)} 
■+Ro \n—2J [w— 2]’-'* {3, 4, (r— 3)} 
.+R';.,[w-2]’-=“[?^-2]» {3, r, (1) } 
+ [%—!]*’■ 
+ [w— 
r+Rr [n~ 2 J [w_2]«{ir+l , ir+2 , (1)} 
[ r even ; or r odd 
J+Rr-^[n-27^->[7^-2]•{K^+l). K^+3), (2)} 
l+Er+''[»»-2]««'>[»-2]"{i(»'+l), i(»'+5), (1)} ; 
and the general term is 
[n-lj Rf[w-2J+t»*-2]'-^'-HHl, H2+5, (r-2S+l-s)}, 
where s extends from s=0 to s—r — 2S, and h from ^ = 0 to or ^(r — 1), according 
as r is even or odd. The expression for the coefficients R“ is 
T?o_W! 
and that of the other coefficients Rf (S= or<l) is not required for the present purpose. 
28. According to a remark already made, the expressions for &c. are at 
once obtained from that for D*"!! by merely writing n — 1, n—2, &c. in the place of n: 
it is however to be noticed, that the quantity ■within the [ ] must not be negative, and 
that on its becoming so, the factorial is to be omitted. 
29. I write now 
MDCCCLIX. 
+s=a, r—2'h—s=(5, 
2 F 
