22 PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



series of curves &U + hV + cW = 0, in involution with the given curves 

 U — 0, V = 0, W = 0, — has its equation expressible in the form 



*JmU-*JIT + J— X nT=0; 

 > p. 



that is, we have the curve T = (the equation whereof contains a variable para- 

 meter) as a zomal of the given trizomal curve JTJj + JmV + JnW = ; and 

 we have thus from the theorem of the decomposition of a tetrazomal deduced 

 the theorem of the variable zomal of a trizomal. The analytical investigation is 

 somewhat simplified by assuming p = ab initio, and it may be as well to repeat 

 it in this form. 



47. Starting, then, with the trizomal curve 



JlU + JmV+ J%W = , 



and writing 



aU + hV + cW + dT = 



as the definition of 7 1 , the coefficients being connected by 



the equation gives 



IU + mV + 2 JEajV- nW =0 ; 



or substituting in this equation for W its value in terms of U, V, T, we have 



(an + cZ) U + (bra + cm) V + 2c JTmUV + dnT = , 



which by the given relation between a, b, c, is converted into 



that is 

 viz., this is 

 or finally 



-^mU- ^lV+2cJl^UV + anT=0; 



&*mU 4- h 2 l V - 2ab JhriJV = — nT } 



■ JmU - b J IV + J^nT= 



48. The result just obtained of course implies that when as above 



a u + \ ) v+eW + aT = o,- + -^+- =0, 



the trizomal curve JlU + JmN + s/nW = can be expressed by means of any 

 three of the four zomals U, V, W, T, and we may at once write down the four 

 forms 



