admit of reduction in the number of Variables. 125 



the evanescence of the 4th, for thus only can the necessary con- 

 formity between the number of affirmative conditions and the 

 number of unimplicated equations come to take effect. The 

 clear and direct putting in evidence of this peculiar species of 

 implication demands and deserves to be minutely considered; 

 and as it must in part borrow its explanation from the very 

 little yet known of syzygetic relations, so it must also throw 

 new light on that great and important, but as yet unformed and 

 scarcely more than nascent theory. 



In conclusion, it is apparent from the demonstration above 



will be zero : the equation expressing this relation is termed a syzygetic 

 equation. 



Thus, if we take the 3 full determinants that can be formed out of the 

 matrix 



a « 



b ^ 

 c y, 

 i.e. ap—bei; 6y— c/3; c«— ay, 



these are in syzygy, for we can form the equation 



c X (a/S- 6a) + a(6y— c/3) + 6(c«»— ay)=0. 



This, however, is not the only equation of the kind that can be formed, for 



y{a^—bcc) + u{by—c^)-i-^{cei—ay)=0 



is also identically true. We see in this case that the evanescence of any 2 

 of the 3 functions a^—bet; by—c^; ccc—a^ will in general imply the 

 third, suhject, however, to special cases of exception. Thus, if the 1st and 

 2nd vanish, the 3rd must vanish unless b and /3 both vanish ; if the 2nd 

 and 3rd vanish, the 1st must vanish unless c and y both vanish ; if the 3rd 

 and 1st vanish, the second will vanish unless a and ct both vanish. It will 

 thus be seen that a peculiar species of astricted syzygy obtains between 

 the 3 proposed functions, which enables us to affirm that in general, and 

 except under extra special conditions, all three must vanish simultaneously. 

 If 2 out of the 3 vanish, and the 3rd does not vanish, it is not merely (as 

 might at the first blush of the theory of syzygy be conjectured) because 

 some one other function vanishes in its place, but necessarily because a, plu- 

 rality of entirely independent functions (2 simple letters as it happens here) 

 each separately vanish. Thus we see how all but one of a set of functions 

 Xi» X2» • • • Xn ™^y *^ general, and yet not universally, necessarily vanish 

 when all the rest vanish : to say that one syzygetic equation such as 



obtains, is not enough to explain the circumstances of the case; the fact is, 

 that several distinct systems of values of x'l, X 2» " -x'n will be found capable 

 of satisfying the equation, so that each of the functions xi> X2> • • • X» will have 

 a system of syzygetic factors attached to it, and these unrelated, in the wide 

 sense, that, if we take x'n, x"n> ^^y two of the syzygetic factors attached 

 to xn, they will not he in syzygy with xi> X2> • • > X^-H so that when 

 these {n — 1) functions vanish, the vanishing of x'« and x"n represents two 

 distinct and completely independent conditions. Thus, in fine, the mutual 

 implication of fimctions will in general denote the possibility of forming a 

 series of syzygetic equations between them, — a remark, this, of no minor 

 importance. 



