357 
411] A MEMOIR ON THE THEORY OF RECIPROCAL SURFACES, 
writing herein 
n' = cl 4- k — cr — 2 6' — 45 — 2\j — 3% — o", 
the value is 
= — 26a + 22« — 206' — 4021 — 20j — 30% — 10a*. 
Substituting for k its value = a (n — 2) + 5 — p — 2o*, we have 
12c' — 26n' = a (22n — 70) - 20C — 1821 — 20j — 30% — 22p — 44o*; 
or substituting for a, p, a their values, this is 
= {n (n - 1) - 26 - 3c} (22™ - 70) - 206' - 185 - 20j - 30% 
— 226 (n — 2) + 44/3 + 667 4- 66£ 
- 27c (n - 2) + 108/3 + 277 4- 270, 
and adding hereto the remaining terms, 
/3 + 26™ - 12c + 2' + 7/ + 8%' - 4- 46" 4- 105' 
-2 -7j -8% -46' -105, 
we have 
/3' = 2™ (n - 2) (lln - 24) 4- b (- 66™ 4-184) + c (- 93™ + 252) 4- 153/3 4- 93 7 4- 66i 
4-6 4- 7/4- 8%' — \& 4- 4(7'4-105' 
- 2 - 27j - 38% 4- *£ 0 - 24G - 285. 
Comparing this with the value of /3', No. 63 of the foregoing Memoir, we should 
have 
0 = is cw - 12c - 12/3 - ^7 
~(x + 1)2' — 14%' — \6' — 96" — 285' 
- (a? 4- 86) 2 + 6? 4- ^% - 2d 4- 285 
4- T Vd (- 165 - 8% - 0 + 165' 4- 8%' 4- 0'), 
or, what is the same thing, 
0 = 13c™ - 48c - 48/3 - 137 4- <t>, 
if for shortness 
0=-(4a;'4- 4)2' — 56%' — 76' — 36C' - 1125' 
-(4a? 4-344) 2 4-24/4-70% —80 4-1125 
4- \g (- 165 - 8% - 0 4-165' 4- 8%' 4- 0'). 
I do not attempt to verify this equation, but I will partially verify a result 
deducible from it; viz. if <!>' is the like function of the accented letters, then we have 
cp _ qy = 11 _ ii', 
where 
n = (4af - bx - 340) 2 + 24j 4- 126% 4- 2245 4- 366' - 0 
-ii/(16.B + 8 x + 0);