PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



25 



thereof, but the equation of any tangent of the conic is ax + by + cz — 0, where 



J my m 



a, b, c are any quantities satisfying the condition — (-— + - = o ; whence, writ- 



a d c 



ing ax + by + cz + &w = 0, we may introduce w — along with any two of the 



original zomals x — 0, y — 0, z = 0, or, instead of them, any three functions of 



the form w ; and then the mere change of #, y, z, m into U, V, W, T gives the 



theorem. But it is as easy to conduct the analysis with ( U, V, W, T) as with 



(#, y, z, w), and, so conducted, it is really the same analysis as that whereby the 



theorem is established ante, No. 47. 



51. It is worth while to exhibit the equation of the curve 



in a form containing three new zomals. Observe that the equation — h t- H — = 



a d c 



is satisfied by a = tyx, b = w»x^» c = ^^ ^ onl y e + 4* + X = ^ ; or say, if 

 9 — a!— a", <p = a"— a, x — a—a"- The equation 



>• J (a- a) (a - a")l U + (a - a") (a' - a)m V + (a" - a) (a" - a')n W 

 + lb J(b-V) (b-byU + (V-W) (V-b)mV+ (b"-b) {b"-V)nW 



+ » J(c-c) (c-c")lU + (c'-c*) (c'-c)mV + (c"-c) (c"-c')nW=0 



is consequently an equation involving three zomals of the proper form ; and we 

 can determine X, m, v in suchwise as to identify this with the original equation 

 *JlU+ JmV + JnW, viz., writing successively U = 0, V = 0, W = 0, we find 



ia'-a") X + (b'-V) (b + {c'-c") , = , 



(a"-a) X + (b"-b) it + (c"-c) v = , 



(a— a) X + (b—V) ib + (c—c' v = , 



equations which are, as they should be, equivalent to two equations only, and 

 which give 



X : /b : » sr 



1,1, 1 





1,1,1 



: 



1,1,1 



b, V, b" 





c, c', c" 





a, a', a!' 



c, c', c" 





a, a, a" 





b, V, b" 



and the equation, with these values of X, n, v substituted therein, is in fact the 

 equation of the trizomal curve *JW + JmV + *Jn W = in terms of three 

 new zomals. It is easy to return to the forms involving one new zomal and any 

 two of the original three zomals. 



Remark as to the Tetrazomal Curve — Art. No. 52. 



52. I return for a moment to the case of the tetrazomal curve, in order to 

 show that there is not, in regard to it in general, any theorem such as that of the 

 variable zomal. Considering the form JTx + >Jmy + *Jnz + Jpw = (the co- 



VOL. XXV. PART I. 



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