406] 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
241 
120(1,1,1,1,1)= (2 m + 77) 5 
+ 10 (2m + iif (- 4m — n — 3a) 
+ 10 (2m + nf (- 32m - 58n + 78a) 
+ 10 (- 4m - n - 3a) (- 32m - 58n + 78a) 
+ 5 (2m + n) (2208m + 2610n - 2358a) 
+ 15 (2vi + n)(— 4m — n— 3a) 2 
- 102912m- 112056n +86760a 
( 31m 5 + 70 m 4 77 + 40m 3 77 2 ' 
— 310777 4 — 460m 3 77 — 120777 2 77 2 
— 235m 3 — 1030m 2 Ti — 400m n 2 
+ 10690m 2 + 16060777/1 +960 t7 2 
+ (- 210m 3 — 180m 2 7i^ 
+ 297Û777 2 + 9 0077771 
— 15630777 — 7 2077 
+ 28440 
^ + a 2 (135777 — 540), ) 
where the correction is 
= — (m — 4) 
which is 
( 31m 4 -186m 3 -979m 2 + 6774m 
+ n(70m 3 — 180t?7 2 — 1750777 + 9060) 
+ 4077 s (m + 3) (m — 2) 
+ a /— 210m 2 + 2130m- 7110' 
\— I8O77 (m — 1) 
+ a - . 13o, 
= — (m — 4) 
( 31 (28 + 3/c) 2 + 110 (28 + 3k) (2t + Si) 
+ (^m 3 +1142m+ 3174 )( K - i) 
+ (- 14777- 63877- 1524 ) (28 + Sk) 
+ (— 39O777 + IIO77 + 4272 ) (2t + 37 ) 
v + (— 210m 2 — 180m77 + 2130777 + 99077 — 7110) *+ 135/c 2 y 
but I have not sought to further reduce this expression, not knowing the proper form 
in which to present it. 
92. The question which ought now to be considered is to determine the corrections 
or supplements which should be applied to the foregoing expressions (a), (a, b), &c., or 
to their equivalents [a], [a] [6] + [a, 6], &c. in order to obtain formulae for the cases 
c. VI. 31