20 PROFESSOR CAYLEY ON POLYZOMAL CURVES, 



or as this is more conveniently written 



( ( J' + g^j) x/tj + jmv + jnwj = b -^f (b s /,rr - c y^r) , 



an equation breaking up into two equations, which may be represented by 



where 



/ - i- a i' /— / a v 



«h - sfl + jQj , V/ 2 - J/ + - d -jj 



Vwi, = Vw - V^gf b ,/» ' %/% = x 7 "' + Jj^// b s 



where, in the expressions for J~l t &c, the signs of the radicals 



JT, Jin, ijn,w . a , -^ • 

 bed / 



may be taken determinately in any way whatever at pleasure ; the only effect 

 of an alteration of sign would in some cases be to interchange the values of 

 {JT V Jm v Jn x ) with those of (Jf v J^~ v Jn )■ The tetrazomal curve thus 

 breaks up into two trizomals. 



44. It is to be noticed that we have 



a n "b 1 "c~a^d 3 /" r d 



m a np 

 + b~ + 6dT 



n a rap 

 + 7 + bd T 



=0 + I)(v+?+t>I> 



that is 



And similarly we have 



ii + ™i + ^ = 

 a b c 



a b c 



The meaning is, that, taking the trizomal curve J I U + l Jm 1 V + J^ W = 0, 

 this regarded as a tetrazomal curve, JIJJ+ Jrn\V + JnJW + *J0T = 0, satis- 

 fies the condition _i- + ^i + ^ i + ^=0; and the like as to the trizomal curve 

 abed 



JLU + Jm7V + JtTW^O. 



