14 REPORT 1875, 



the other columns. It is evident in this example that the three cycles include all 

 the sets which contain a sequence of two and one break ; so that the theorem is 

 proved for this case. A little consideration, however, shows too that if aU the sets 

 formed according to any fixed rule regulating the sequences and breaks be written 

 down, they must consist of a group (or cycle), or of an aggregation of several groups 

 each of which is rational. For consider any one set : it must belong to a group, for 

 we can obtain a group from it by increasing the numerals in it, each by unity, succes- 

 sively tiU it reproduces itself ; also no set can be common to two groups. 



We thus see that the truth of the general theorem depends upon two considera- 

 tions, viz. (i) upon the remark that any function such as 



■where 



(plx, x) + (f){x, X^') + <f)(x, X^)+ . . . (f)(x, Xn), 



4>{x, s) = A+B3+0s' . . . +P3"-i 



(B, C . . . P being any non-infinite functions of x), is rational ; and (ii) upon the 

 proposition that the total series of sets formed by arranging the numerals according 

 to any law of sequences and breaks consists of the aggregation of groups. 



It is evident that the theorem is equally true if we imderstahd 1, 2, 3 ... to mean 



(Y-ccy, {i-x^r, (1-xy-,..., 



or even 



(l-.r)", Q.-x'^Y, (l-.r')» . . . , 



or, in fact, any functions 



^{x, x), <l>{x, X-) . . . , 



subject to the conditions that ^(.i', 1) = 0, and that in the development of the type 

 expression involving s the coefficients of the terms in s", s», s^n . . . are to be inde- 

 pendent of X. 



If the former condition is not satisfied, the theorem is still ti'ue if the sets are 

 formed from the series of numerals 1, 2, 3 . . . n (i. e. including n). With this 

 alteration therefore the theorem is true for 



1+x, l+a:^ . . . or for (l-1-.r)", (l+^O" • • • 



The point of the theorem lies in the fact that functions of the roots of an equation 

 which are not in appearance symmetrical, are rational ; but it is generally quite 

 easy to go further and assign the absolute values of any of the expressions con- 

 sidered, since the value of any gi-oup is readily assigned : ex. gr. consider the 

 group written above, viz. 124+23o-f 346-J-56] j this is 



2(1— ^)(1— a:2)(l-a;=3) = 2 (l+As+Bs^+Cs^), for s=.r,a?=' . . . x' 



= 7, 



and generally each group =n (1, 2 . . . standing for 1— .r, 1 — x^ • • • )> ^° ^^^* ^^® 

 value of an expression = n times the number of groups it contains. 



It is also to be noted that very often it is not necessary that x should be a prime 

 7ith. root ; ex. gr. in the expression just written it is enough that 



22=0, 2s2=0, 2s' = 0, 



i. e. that neither x, x'^, nor x' should be unity. In this particular instance, since 7 

 is prime, every root is a prime root; but whatever 7i may be, if there be only 

 three numerals in each set, it is enough that a; is an nih. root which is not a square 

 or cube root of unity ; and the generalization of this remark is obvious. 



On some Geometrical Theorems, By W. Hatben. 



The paper was principally concerned with the properties of an isosceles triangle 

 in which the squares on the equal sides are each double the square on the unequal 

 side obtained geometrically : (1) this triangle can be constructed without the use 



