134 



sides, there is but one form of relation : if, for instance, in the equation 

 [12] = 0, which is the condition for a proper dodecagon, the function 

 [12] could be decomposed into rational factors; then equating each 

 of these factors to zero, we should have so many distinct forms of re- 

 lation for a proper dodecagon. I believe that the assumption and 

 reasoning are valid ; but without entering further into this, I take it 

 for granted that in the general case the functions [3], [4], &c. are in 

 fact prime. But the coefficients /3, y, 8, or b, c t d, instead of being 

 so many independent arbitrary quantities, may be given as rational 

 functions of other quantities (if, for instance, the two conies are cir- 

 cles, radii R, r, and distance between the centres a, then /3, y, 3 will 

 be functions of R, r, a) : and it is in a case of this kind quite con- 

 ceivable that the functions [3], [4], &c., considered as functions of 

 these new elements, should cease to be prime functions. In fact, in 

 the case just referred to of the two circles (the original case of the 

 Porism as considered by Fuss), the functions [4], [6], &c., which 

 correspond to a polygon of an even number of sides, appear to be each 

 of them decomposable into two factors : the memoir contains some 

 remarks tending to show a priori that in the case in question this 

 decomposition takes place. I was led to examine the point by the 

 elegant formulae obtained in an essentially different manner by M. 

 Mention, Bull, de 1'Acad. de St. Pe't. t. i. pp. 15, 30 and 507 (1860), 

 in reference to the case of the two circles (it thereby appears that the 

 decomposition takes place for the quadrangle and the hexagon) j and 

 these formulae are reproduced in the memoir. 



II. " On a New Auxiliary Equation in the Theory of Equations 

 of the Fifth Order." By ARTHUR CAYLEY, Esq., F.R.S. 

 Received February 20, 1861. 



(Abstract.) 



Considering the equation of the fifth order, or quintic equation, 

 (*Jt>, l) 5 =(vx l ) ( a? 2 ) (vx 3 ) (v-xj (vxj = 0, and putting 

 as usual i A>==# 1 + w# 2 + (o 2 # 3 + w 3 # 4 +w 4 < z> 5 , where w is an imaginary 

 fifth root of unity, then, according to Lagrange's general theory for the 

 solution of equations, /iu is the root of an equation of the order 24, 

 called the Resolvent Equation, but the solution whereof depends 



