SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 1065 



tbe transformation to secondary axes is effected by substituting in every term the values 

 y/l{x-y) for X and */% {x + y) for Y, and expanding. Let A p {X n ~ p Y p + X p Y n ~ p } be any 

 pair of homologous terms ; their equivalent in the transformed equation is 



(^A P {(x-yr-P{x + yy> + (x- y )p(x + yy-P\ . . . (4). 



Expanding the first term within the bracket in columns, the coefficients are — 



i+ j, +£ ^^i + P .(P-i)(y-2) +&c . 



f) (p — 1) 



— (n— p) — (n— p)xp — (n— p)x- i. 9 — - — &c. 



(n — p)(n — p — 1) (n — p)(n — p — 1) , „ ,_. 



+ 12 + 12 x P+ kc ( 5 )« 



The coefficients of the expansion of the second term within the brackets are — 



i_ „ y(j?-l) p.(p-l)(p-2) 



1 -i- 12 1.2.3 "*" ' ' • ■ ■ w 



P (p — 1) 

 + (n— p) — (n— p)xp + {n— p)x 1 ~ * ~ — 



, (n-p)(n-p-l) (n-p)(n-p-l) ^^ , 

 ■+■ 1 2 1.2 x ^" 1 " ' 



where in the first set of terms the sign x is used to separate the factors derived from the 

 expansion of (x—y) n ~ p from those derived from the expansion (x + y) p , and similarly in 

 the second set of terms. 



The quantity in the first column (always unity) is the coefficient of x n in the trans- 

 formed equation. The sum of the quantities in the second column is the coefficient of 

 x n_1 y in the transformed equation, and so on. We see that the expansion of the second 

 term of the pair is the same as that of the first term, except that in the second, fourth, 

 and every other alternate column, the signs + and — are interchanged, and therefore 

 the sums of these columns in the two expansions is zero. If the pair of homologous 

 terms have contrary signs, then in the expansion the sum of the first, third, &c, columns 

 is zero. From this analysis is derived the following abstract of results, hereafter referred 

 to as the Rule of Signs. 



1. From the mode of formation of the transformed equation it is always symmetrical 

 if the original equation is symmetrical. 



2. If in the original equation the terms constituting a symmetrical pair are of even 

 degree and have like signs, i.e., both positive or both negative, then in the expansion of 

 these terms in the transformed equation the sum of the partial coefficients is zero for all 

 terms of uneven potvers. 



3. If in the original equation the terms constituting a symmetrical pair are of even 



