14 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



30. It has been mentioned that the form ^//(e + L<b) is considered as includ- 

 ing the form JIq + Z$, that is, when / = 0, the form JTq>. If in the equa- 

 tion of the p-zomal curve there is any such term — for instance, if the equation 

 he *jLq> + */m(0 + M<&) + &c. = — the radical JT^ contains the factor 

 <£* ; but if L contains as factor an odd or an even power of <J>, then J17q> will 

 contain the factor <£« where a is either an integer, or an integer + i . Consider 

 the polyzome JZ<& + s/m(Q + Mq>) + &c, belonging to any particular branch 

 of the curve; the radical JLq contains, as just mentioned, the factor <£«, and if 

 the remaining terms Jm(Q + J/<J>) + &c., are such that the expansion contains 

 as factor the same or any higher power of <£, then the expansion of the polyzome 

 JL$>+ s/m(Q + M$>) + &c., belonging to the particular branch will contain the 

 factor <$« ; and similarly we may have branches containing the factors <l> a , <££, &c, 

 whence, as before, if w = a + /3 + &c, the Norm will contain the factor $> a ; the 

 only difference is, that now a, (3, &c, instead of being of necessity all integers, 

 are each of them an integer, or an integer + \ ; of course, in the latter case the 

 integer may be zero, or the index be = \. It is clear that w must be an integer, 

 and it is, in fact, easy to see that the fractional indices occur in pairs; for 

 observe that a being fractional, the expansion of »Jm(0 + M<P) + &c, will con- 

 tain not <j> a , but a higher power, 4>« + 1, where a + q is an integer ; whence each 

 of the polyzomes JZ<b± {Jm(Q + J/<£) + &c.) will contain the factor <$>*. 



31. Observe that in every case the factor $ a presents itself as a factor of the 

 expansion of the polyzome corresponding to a particular branch of the curve; 

 the polyzome itself does not contain the factor $ a , and we cannot in anywise say 

 that the corresponding branch contains as factor the curve c}> a = ; but we may, 

 with great propriety of expression, say that the branch ideally contains the curve 

 c£> a = 0; and this being so, the general theorem is, that if we have branches 

 ideally containing the curves <£« = 0, <E>£ = 0, &c. respectively, then the »/-zomal 

 curve contains not ideally but actually the factor <i> u = (a> = « + (3 + &c.), the 

 order of the y-zomal being thus reduced from 2"- 2 >* to 2>~ 2 r — w{r — s); and 

 conversely, that any such reduction in the order of the ^-zomal arises from factors 

 cjy* = 0, & 1 * = 0, &c, ideally contained in the several branches of the v-zomal. 



32. It is worth while to explain the notion of an ideal factor somewhat more 

 generally ; an irrational function, taking the irrationalities thereof in a deter- 

 minate manner, may be such that, as well the function itself as all its differential 

 co-efficients up to the order a — 1, vanish when a certain parameter $ contained 

 in the function is put = ; this is only saying, in other words, that the function 

 expanded in ascending powers of $ contains no power lower than $« ; and, in 

 this case, we say that the irrational function contains ideally the factor $ a . The 

 rationalised expression, or Norm, in virtue of the irrational function (taken deter- 

 minately as above) thus ideally containing $ a , will actually contain the factor 



