100 
Again, 
BIE. W. SPOTTISWOODE ON THE CALCULHS OE SYMBOLS. 
4'iT4- ■4'o^ + ‘Po 4 ;^ 
p2Y‘^-{-p2^‘^+p2Y 
Vl Vl Vl 
Hence the condition that 4'i(f)‘^+4'o(f) i^iay be an internal factor of ?> 2 (^)^+<Pi(f)^+?’o(i’) 
will be 
'^o(g) 
^l(g) 
'^l(g) 
= 0 . 
( 2 .) 
Before proceeding further, we may remark that the remainder, after internally dividing 
P 3 {§)'^-\-P 2 {§)'^^-{-Pi{§)'^-\-Po{s) by '4'i(f)‘^+'4'o(f)? can differ from that last above found 
only in respect of the remainder arising from the division of P 3 {§)‘r^ by the factor in 
question; hence we have now only to divide the term p3{§)'7^ by '4'i(f)^+'4'o(f)i 
the remainder so found to (2,), in order to have the condition required for the third 
degree. Proceeding to the division, writing %= and omitting for the present (p 3 , which, 
since the division is internal, can be replaced as an external factor in the remainder, we 
have 
—Xjr‘^—2-)C‘r—x 
—X'^‘'—X^^—XX 
{x^—^x}'^+xx—x 
Xx‘—^x)‘^+x"—^xx 
-x'+^xx-x- 
Hence the condition that 4'i(s)'^~i"4'o(^) be an internal factor of ?' 3 (^)'^+?> 2 (f)’r^ 
-hPi(s)^+PoQ) will be 
PoQ)-piQ) 
M§1 
Pi(§) 
0 , (3.) 
the identity of which with Mr. Bussell’s condition, given in p. 75 of his paper, I have 
verified. 
