118 
MR. W. SPOTTISWOODE ON THE CALCULUS OF STIVIBOLS. 
and in this the coefficient of is ( — )' X 
0 . . 0 0 
: ..0 0 
0 
0 
0 
0 
0 . . — 1) 0 
0 . . 0 ■4/l(7r+^') 
0 
0 
0 
0 
0 .. 0 0 .. ^p,(^+N-2) ^|/„(^+N-l) 
0 .. 0 0 .. 0 4/j(^4-N-l) ' 
— l)4'i(’*'+*) • • •4/,(^r+N — 2)4 /,(t+N— 1) ; 
whence the whole expression 
— v*-N/_ Vrn / , + ^ — 1) 
"i=:0 V JyA 1) .. 4'i(7I' + « — !)’ 
which agrees with the result given in the Philosophical Transactions, vol. cli. p. 73. 
In the particular case of N = 4, M=2, the final remainder in internal division is 
^ 'la 1 ) I'a ~ 2 ) vf/g (?>■ — 3 ) 
<PiM 
4^iW 
0 
0 
4/2(t— 1) 
4/o(^— 1) 
<Ps(^-2) 
0 
(p,(^—3) 
0 
0 
yp^(‘r—3) 
, 1 
(PoM 
4'o(^) 
0 
0 
~ 2)l2(’^ “ 3)l2(’f — 4) 
<P2(t— 2) 
4^2(’J'— 2) 
n^-2) 
^o(^-2) 
3) 
0 
\l/i(7r— 3) 
(P4(9r— 4) 
0 
0 
yp^(7r—4); 
and in external division it is 
1 
^ + 2 ) 
+ 
1 
4'.2(^)4'2(^+ l)'p2('^+ 2) 
<PiW 
'f^l( 75 -) 
0 
0 
<P2W 
4^2(75-) 
4^1(75- + !) 
4 'o( 7 !- + 2) 
13(7^) 
0 
4^2(75-+!) 
4 'i( 75 - + 2) 
14(75-) 
0 
0 
4^2(75-+ 2 ) 
lo( 7 i-) 
4^0(75-) 
0 
0 
12 ( 7 !-) 
4^2(75-) 
4^1(75-+!) 
4^0(75- + 2 ) 
13(75-) 
0 
4 /a( 7 r+l) 
4'i(75-+1) 
14(75-) 
0 
0 
4 ' 2 ( 7 r). 
