116 
PBOCEEDINGS OF SOCIETIES. 
but the expressions for the remaining quantities, viz., b', p , o', 4/3' + 7 ', 
8 ic' - 18 k! would he very complicated. If we suppose that b, c, 4 / 3 + 7 , 
8 k - 18 h, vanish, or, what is the same thing, attend only to the terms 
which contain n alone, we have— 
2 V - n(n - 1 ) (ft - 2 ) (ft 3 - ft 2 + n - 12 ), 
p'= n(n - 2 ) (ft 3 - n 2 + n - 12 ), 
a = 4 n(n - 2), 
4/3' + 7 ' = 4 n\n - 2) ( n 3 - 3ft 2 + 3ft - 3), 
8 # - 18# = n(n - 2) (ft 10 - 6 ft 9 + 16ft 8 - 54ft 7 + 164ft 9 - 288# + 403# 
- 482# + 348# - 242^ + 60), 
which agree with the values which Mr. Salmon has obtained for (3', 7 ', 
h', k' by means of the twenty-one equations, and the additional equa¬ 
tions (peculiar to the case in question, of a surface of the degree n with¬ 
out singularities, and which are obtained by him from independent con¬ 
siderations), i' - 0, and /3' = 2ft(ft - 2) (11ft - 24). 
The system of seventeen equations completely accounts for the re¬ 
duction of the order of the given surface considered as the reciprocal of 
the reciprocal surface, hut the omitted equations are important for 
other purposes. "We may by means of them express i, t in terms of the 
data for the system of twenty-one equations, viz., ft, b, c, (3, 7 , h, k; 
and, effecting this, and annexing the corresponding values of i f , t r , we 
have the supplementary system— 
4 i = ( 5 ft - 6 )<? - 6 c a + 12 Ji- 57 , 
24* = - ( 8 ft - 8)b + (15ft - 18> + 2(4 b 2 - 9c*) -I6k + 3 6h + 20/3-15 7 , 
4 *'= (5ft' - 6 )c' - 6c 2 +12# - 57 ', 
24*'= - ( 8 ft'- 8)b' + (15ft'- 18c') + 2(4 b' 2 - 9 c' 2 ) -16# + 36#+ 20/3' -15 7 , 
to which I annex also, without transformation, the four equations for 
q, r, q ', r', viz. 
q = b* - b - 2k - 3^ - 6 *, 
r = c 2 - c - 2h - 3/3, 
q — V 2 - # - 2 # - 87 ' - 6 *', 
r’ = c' 2 - c'-2h'~ 3/3. 
The last two of which, neglecting singularities, give— 
q’ = 4ft(ft - 2) (ft - 3) (ft 2 + 2ft - 4), 
r' = 2ft(ft - 2) (3ft - 4), 
which are the values given by Mr. Salmon. I remark, in conclusion, 
that there is considerable difficulty in the geometrical conception of the 
points i and the planes i', and the subject appears to. require further exami¬ 
nation. In the case of a surface of the order n without multiple lines, 
we have not only i— 0 (which is a matter of course), but also - 0 . In 
my paper before referred to, I showed, or attempted to show, by geo- 
