PROFESSOR CAYLEY ON POLYZOMAL CURVES. 89 



pair symmetrically situate in regard to the axis. Hence also dps. = 5; class = 10; 

 deficiency = 1. 



And there is apparently a seventh case, which, however, I exclude from the 

 present investigation, viz., this would be if we had 



( 1 ,. 1 , 1 , 1 ; )( Jl J.^ Jn, Jp)=*-0, 



a , b , c , d , 



a 2 , b 2 , <? , d 2 , 



a" 2 , b" 2 , c" 2 , d" 2 , 



that is, a, b, c, d denoting as before, if we had 



Jl : Jm : Jn : Jp = a : b : c : d, and aa" 2 + bb" 2 + cc" 2 + dd" 2 = . 

 For observe that in this case we have 



I 

 a 



that is, the supposition in question belongs to the decomposable case. 



a A° + bB° + cC° + dD° = , and Z+™+* + P = 0; 



abed 



The Decomposable Curve — Art. No. 197. 

 197. We have next to consider the decomposable case, viz., when we have 



aA° + bB° f cC° + dD° = ; 



see ante, Nos. 87 et seq. — it there appears that (unless the centres A, B, C, D 

 are in a line) the condition signifies that the four circles have a common ortho- 

 tomic circle ; and when we have also 



U™ + l + l = o. 



abed 



The formulae for the decomposition are given ante, Nos. 42 eo seq. Writing 

 therein A°, B°, C°, D c in place of U, V, W, T respectively, it thereby appears 

 that the tetrazomal curve JlIK 7 + s/mB° + */nC° + JpU" = 0> breaks up into 

 the two trizomal curves 



where 



Jl^ + Jm x & + J^C = , Jl 2 A° + J m , 2 Q° + Jn 2 C° = , 



Jm, - Jm - J* Sbjn, J^ = J m + J± f b J 



A = Jn, + V^f c J m , Jn 2 = Jn - J ^ f o Jm , 

 and where we have 



h + m 1 + n 1 = t k + 012 + ^ = . 



a b c a b c 



VOL. XXV. PART I. 



