206 Mr. A. Cayley on Theorems relating to the 



John Leslie and others have been perplexed by the varying indi- 

 cations of their instruments on days equally bright ; but all these 

 anomalies are completely accounted for by reference to this 

 newly-discovered property of transparent aqueous vapour. Its 

 presence would check the earth's loss ; its absence, without 

 sensibly altering the transparency of the air, would open wide 

 a door for the escape of the earth's heat into infinitude. 



XXVIII. Theorems relating to the Canonic Roots of a Binary 

 Quantic of an Odd Order. By A. Cayley, Esq.* 



I CALL to mind Professor Sylvester's theory of the canonical 

 form of a binary quantic of an odd order; viz., the quantic 

 of the order 2n -f 1 may be expressed as a sum of n + 1 (2n + l)th 

 powers, the roots of which, or say the canonic roots of the quantic, 

 are to constant multipliers pres the factors of a certain covariant 

 derivative of the order (»+l), called the Canonizant. If, to fix 

 the ideas, we take a quintic function, then we may write 



(a, b, c, d } e,fXx>y) 5 = A (te + m/Y+ A. ! (l'x + m'y) 5 + A"(Z"#+m'V) 5 

 (it would be allowable to put the coefficients A each equal to 

 unity ; but there is a convenience in retaining them, and in con- 

 sidering that a canonic root lx + my is only given as regards the 

 ratio / : m, but that the coefficients /, m remain indeterminate) ; 

 and then the canonic roots (Ix + my), &c. are the factors of the 

 Canonizant 



y 3 , -y*x, yx*, -a* 



a , b , c , d 

 b , c , d , e 

 c , d , e , f 

 It is to be observed that this reduction of the quantic to its 

 canonical form, i. e. to a sum of ra+1 (2rc+l)th powers, is a 

 unique one, and that the quantic cannot be in any other manner 

 a sum of 7i + 1 (2n + 1 ) th powers. 



Prof. Sylvester communicated to me, under a slightly less 

 general form, and has permitted me to publish the following 

 theorems : — 



1. If the second emanant {Td x + Yc^) 2 U has in common with 

 the quantic U a single canonic root, then all the canonic roots 

 of the emanant are canonic roots of the quantic ; and, moreover, 

 if the remaining canonic root of the quantic be rx + sy, then 

 (X, Y), the facients of emanation, are = (s, —r), or, what is the 

 same thing, they are given by the equation 



canont. U(X, Y in place of x, y) = 0. 

 * Communicated by the Author. 



