of Edinburgh, Session 1876 - 77 . 387 
and now using (a), to express x 0 i n terms of 4>, we find 
'I' (w) = ty(n- l)424>(ft- 2)4- 3 {444 - 3)4- 4 4/(3)} 
4 (w -1)4 2 *(w - 2) 4- 3 ty(n -3)4 4#(ft - 4) 
+ 5{f(n - 5)4- 4 4/(3)} 
4^(n - l)4 24(n - 2) 4 3^(ft - 3) + 4 *(n - 4) 
4 5{4/(ft - 5) 4 4^(3)} 
4 V(n - 1) 4 2&(n - 2) 4 3 {^(ft - 3) 4 4 ^(3j) 
where, on the first line the coefficient of the third and all the fol- 
lowing terms is 3, on the second line the coefficient of the fifth and 
all the following terms is 5, on the third line the coefficient of the 
seventh and all the following terms is 7, and so on, the middle term 
(when such occurs) having a 1 superadded. 
Hence, for the determination of 4/(ft) when 4>(ft - 1), - 2), ... 
are known, we have 
^(ft) = (n-2)^(w-l)4(2w- 4)*(w-2)4(3ft- 6)*(n-3) 
4 (4n - 10)^(ft — 4)4 (5ft — 14)\k(ft — 6) 
4 (6ft - 20)^(ft - 6) 4 (7ft - 26>k(ft - 7 ) 
+ 2 
where the coefficients proceed for two terms with the common dif- 
ference ft -2, for the next two terms with the common difference 
ft - 4, for the next two terms with the common difference n - 6, and 
so on. 
And as it is self-evident that 4/(2) = 0, we obtain 
*(3) = 14/(2)41 - 1 
4/(4) = 24/(3) = 2 
4/(5) = 3^(4) 4 64/(3) 4 1 =13 
4/(6) = 44/(5) 4 8^(4)412^3) = 80 
^(7) = 5' v ff(6) 4 10¥(5) 4 15^(4) 4 1 8^(3) + 1 =579 
4/(8) = 64/(7) 4 12^(6) 4 18¥(5) 4 22^(4) 4 26^(3) = 4738 
and so forth. 
