152 Proceedings of Royal Society of Edinburgh. [sess. 
They will be found to be formulae really established by the 
investigation to which we now proceed. 
D. General Investigation . 
We assume that (1) to (4), which have been established in 
fundamental cases, hold when the total number of vectors (m + n) 
involved does not exceed some odd numbers (2cr+l). First, we 
will show that they still hold when (m + n) does not exceed (2 <j + 2) : 
next when it does not exceed (2o- + 3). That is to say, assuming 
formulae (1) to (4) as they are, we have first to find S(2r+2, 2 s), 
S(2 r, 2 8 + 2), S(2 r.+ 1, 2s + 1), &c., and then S(2 r + 1, 2s + 2), 
S(2r + 2, 2s + 1), &c. If the same law is found to hold, the 
formulae will have been completely established. 
We have, 
S(2r, 2s + 2) = S. a 1 B'(2r - 1, 2s + 2) + S. ai B'(2s + 1, 2 r) 
= {C(r — 1, s+ 1) 4- C(s,r)}Sa 1 a 2 ^ 
l=C(r, s + l)S(K 1 a 2 je ........ (5) 
So too, S(2r + 2, 2s) = 8(2 s, 2r + 2) = C(s,r + l)Sa lV ; . . (6) 
Z(2r, 2s + 2) = S . a x Y'(2r - 1, 2s + 2) + 8. a 1 Y / (2s + 1, 2r) ; 
or, Z(2r, 2s + 2) = 0 . ...... (7) 
so too, Z(2r + 2, 2s) = 0 ; . ...... (8) 
S(2r+ 1, 2s + 1) = S.a 1 B'(2r, 2s + 1) - S ai B'(2s,2r + 1); 
or, S(2r+1, 2s+ l) = 0; ...... (9) 
Z(2r+ 1, 2s + 1) = Sa 1 Y'(2r, 2s + 1) - Sa^+g .... a 2r+2s+2 V . . . a &+1 + . . 
= Sa 1 Y'(2r, 2s+ 1) - S . a x a 2 . . . a 2s+1 Ya 2s+2 . . . . 4 - . . 
= S.a 1 Y'(2r, 2s + 1) - S. ai Y'(2s, 2r+ 1) 
= {C(r, s) - C(s, rJJSojo^ ; 
or, Z(2r+ 1, 2s-f 1) = 0; (10) 
assume, 
Y(2r,2s+2)«Y 1 + V a + Y 8 + Y 4 , 
