108 
ME. W. SPOTTISWOODE ON THE CALCULUS OF STJMBOLS. 
(N-M-P-?) 
The last remainder is 
i. e. ^=0, m=l, or^=l, m=0, 
^^’n+p . 
whence for 
we have 
q=0 q—1 q=2, 
2 >= 0 , [ 2 , 0 ]= 1 , [ 1 , 0 ] = 1 , [ 0 , 0 ]= 1 , 
^=1, [2, 1]=2, [1, 1]=1, [0, 1]=0. 
Hence for m=l the above expression gives 
and for m=0 it gives 
Again, for 7n+q)=0, i. e. q)=0, m=0, we have 
Hence the total remainder is 
It may be useful to compare these results with the actual division, in the above 
example. 
4/27r" + 4/,7r -h \}/o^ + <P3’r' + <p27r" + <p iTT 4- <po Tt" + -^ TT + ^ 
+ 2<P4 'p tt" + (P4 ^ tt" 
^2 V2 
+ <P4^’r"4-2<P4^5r'*4-<p4^7r 
'4^2 
+ <P,^7r^-\-2<p,^7r + <p,';^ 
41 
4^2 
41 
'4^2 
4^2 J 
0,7r® + 
4'^ 2 
TT 
4'2 
4^2 
+4>.|eT+ 
T 2 
(X) 
'4^J 
^27r''+;j,-02\}/,7r 
+ X ^0 
which agrees with the result before found. 
