86 PROFESSOR CAYLEY ON THE THEORY OF RECIPROCAL SURFACES. 
and at the same time introducing the terms in co, and writing down also the terms in 6 
as they stand, we have 
4 i = . 
. . • — 2^+ 30 — 3(y, 
2tt = . 
. . — 6^ + 90 — 9^, 
2(7= . 
. . — 0— CO, 
8§ = . 
. • + 5^— 90+ 9 co. 
8k = . 
. . — 6^ + 170 — 25 co. 
2h = . 
. . + 6^ — 90+ 15a;, 
8 n' = . 
.. -30 X + 210 -45a;, 
d — . 
. . -12* + 100- 20a;. 
The equations of No. 11, used afterwards, No. 53, should thus be 
M + 6r= ( 5n— 12)c— 180— 5y— 2^+30 _ 8><o, 
— 2it— 8^+18r=( — 8w+16)6+(15w— 36)c— 340+9y+4? — 6%+90— 9a ; ; 
and from these I deduce 
Uq+^ r= (M n -88)b+(^n-63)c-^p-^y-132t-87i-22j-\±x+-<r 0 - 
6. In No. 32 we have (without alteration) 0 = 16 ; but in the application (Nos. 40 and 
41) to the surface FP 2 + GR 2 Q ;i =0 we have 0=0, and there are co—fyq off-points, F=0, 
P = 0, Q — 0, and y^—gpq close-points, G = 0, P=0, Q=0. The new equations involving 
co are thus satisfied. „ 
7. I have ascertained that the value of $ obtained, Nos. 51 to 64 of the memoir, is 
inconsistent with that obtained in the “Addition” by consideration of the deficiency, 
and that it is in fact incorrect. The reason is that, although, as stated No. 53, the 
values of two of the coefficients D, E may be assumed at pleasure, they cannot, in con- 
junction with a given system of values of A, B, C, be thus assumed at pleasure ; viz. A, 
B, C being =110, 272, 44 respectively, the values of D, E are really determinate. I 
have no direct investigation, but by working back from the formula in the Addition I 
find that we must have E = 315 ; the values of the remaining coefficients then are 
F=- 2 -, G=— 4* H=-iop, 1= — 198 ; 
or the formula is 
P'=2n(n-2)(lln-24) 
-(110^-272)6 + 442 
-(^n-315)c+^r 
+4 A 0+ J ^P++1986 
— hC — gB —xi —Xj — — vO —fco 
- h’C 1 -g'B'-x'i'-X'j 1 - ^ - v'ff -/V ; 
but I have not as yet any means of determining the coefficients f, f of the terms 
in co, cJ . 
