It is scarcely necessary to remark that from (6) we at once derive a formula, however, which for obvious reasons cannot often admit of useful application to definite integrals. The elliptic-function identities that are given in the Fundamenta Nova would, as a rule, be very difficult to verify in an elementary manner; and the facility with which results of a purely algebraical character, but which seem to transcend the ordinary methods of algebra, are established affords one of the most obvious illustrations of the great power of elliptic functions as a branch of analysis. Of course some of the identities, such as those on p. 103 of the Fundamenta Nova, are very easy to prove by mere expansion; but there is a very interesting one of the more difficult kind which was verified by Gauss. This verification is to be found under the title Zur Theorie der neuen Transscendenten,' Werke, vol. iii. p. 446; and the paper containing it consists of a collection of seventy-three formulæ relating to elliptic functions (or, as they are there described, relating to the arithmetico-geometric mean), with here and there a few words of directions, and is rather the materials for a memoir than a memoir itself. It was taken from a note-book of Gauss's, "finished April 28, 1809," and first published after his death among his collected works in 1866. At the bottom of p. 447, immediately following fourteen formulæ of a different kind, there occur the following: "Anderer Beweis "15. (1-2x+2x1—...) (1 + 2x+2x1+...) = (1-2x2+2x3-...). "16. (1−2x+2xa— ... )2 + (1+2x+2x1+...)2 =2(1+2x2+2x+...)2. “17. (1+2x+2x2+.....) (x3+x2+..... )=(x*+xk+.....)2. “18. (1+2x+2xa + . . . )2 + (2x* +2x2+.......)2 = (1+2x1+2x2+.....)2. “19. (1+2x+2*+...)=(1−2+2 *-...) + (2x+2x2+...)4.” But for the word Beweis at the beginning, it would appear as if this was merely a list of five equations allied to one another in form; and even with the heading it is not easy to see at once what the property proved is, or how the proof is effected. The identity, however, the mode of demonstration of which is sketched in these formulæ, is the elegant theorem 19, which results from substituting for k and k' in terms of q in k2+k12=1 ; and the four formulæ 15-18 are subsidiary results required in the process. As an algebraical demonstration of 19 is valuable, I proceed to expand the proof the steps of which are thus briefly indicated by Gauss in the above extract. The truth of 16 is very readily seen; for all the terms on the left-hand side involving an exponent which is the sum of an even and an uneven square vanish, while the sum of two even squares 4m2 + 4n2=2{(m+n)2+(m—n)2}, and the sum of two uneven squares (2m+1)2 + (2n+1)2=2 { (m+n+1)2+(m—n)2}, which together give the doubles of the sums of all pairs of squares; and the accuracy of the coefficients is evident on consi deration. Formula 16 being thus established, for brevity write a= 1+2x+2x16+..., B=2x+2x+2x25 + ...; and let a, and B, represent respectively a and B with a written for x; then 16 is or (a−B)2 + (a + B)2=2(a1+B1)2, (A) a2+B2= (α, +B1)2. Herein write a-1 for x, and B2 becomes -2, so that which, writing a2 — ß2 as (a—B) (a+ß), is 15. By addition of (A) and (B), a2=a2+622, which, on writing a for x, is 18. We are now in the position to prove 19; for = {(a+B)2+(a— B)2 } { (a + B) 2 — (x — B)2 } by (A) =8(α, +B1)2aß 25 ..}, on substituting for a,, B, their values, and using 17 with a4 written for x, which transforms aß into the second squared factor, by applying 17 again, 2 being therein written for x; and the formula 19 is thus established by elementary algebra in a most simple and elegant manner. The formula 18 is not required in the proof, but was no doubt added by Gauss for the sake of completeness. I CONSIDER a "diagram," viz. a set of points H, O, N, C, &c. (any number of each), connected by links into a single assemblage under the condition that through each H there passes not more than one link, through each O not more than two links, through each N not more than three links, through each C not more than four links. Of course through every point there passes at least one link, or the points would not be connected into a single assemblage. In such a diagram each point having its full number of links is saturate, or nilvalent: in particular each point H is saturate. A point not having its full number of links is univalent, bivalent, or trivalent, according as it wants one, two, or three of its full number of links. If every point is saturate the diagram is saturate, or nilvalent; or, say, it is a "plerogram;" but if the diagram is susceptible of n more links, then it is n-valent; viz. the valency of the diagram is the sum of the valencies of the component points. Since each H is connected by a single link (and therefore to * Communicated by the Author. a point O, C, &c. as the case may be, but not to another point H), we may without breaking up the diagram remove all the points H with the links belonging to them, and thus obtain a diagram without any points H: such a diagram may be termed a "kenogram:" the valency is obviously that of the original diagram plus the number of removed H's. If from a kenogram we remove every point O, C, &c. connected with the rest of the diagram by a single link only (each with the link belonging to it), and so on indefinitely as long as the process is practicable, we arrive at last at a diagram in which every point O, C, &c. is connected with the rest of the diagram by two links at least this may be called a "mere kenogram." Each or any point of a mere kenogram may be made the origin of a "ramification;" viz. we have here links branching out from the original point, and then again from the derived points, and so on any number of times, and never again uniting. We can thus from the mere kenogram obtain (in an infinite variety of ways) a diagram. The diagram completely determines the mere kenogram; and consequently two diagrams cannot be identical unless they have the same mere kenogram. Observe that the mere kenogram may evanesce altogether; viz. this will be the case if the diagram or kenogram is a simple ramification. A ramification of n points C is (2n+2) valent: in fact this is so in the most simple case n=1; and admitting it to be true for any value of n, it is at once seen to be true for the next succeeding value. But no kenogram of points C is so much as (2n+2)-valent; for instance, 3 points C linked into a triangle, instead of being 8-valent are only 6-valent. We have therefore plerograms of n points C and 2n+2 points H, say plerograms CH2n+2; and in any such plerogram the kenogram is of necessity a ramification of n points C; viz. the different cases of such ramifications are* The distinction in the diagrams of asterisks and dots is to be in the first instance disregarded; it is mad ein reference to what follows, the explanation as to the allotrious points. n=5. + n=6. (a) (8) (7) (a) (B) (2) (8) where the mathematical question of the determination of such forms belongs to the class of questions considered in my paper "On the Theory of the Analytical Forms called Trees," Phil. Mag. vol. xiii. (1857) and vol. xv. (1859), and in some papers on Partitions in the same Journal. The different forms of univalent diagrams C" H2n+1 are obtained from the same ramifications by adding to each of them all but one of the 2n+2 points H; that is, by adding to each point C except one its full number of points H, and to the excepted point one less than the full number of points H. The excepted point C must therefore be univalent at least; viz. it cannot be a saturate point, which presents itself for example in the diagrams n=5(y) and n=6(8). And in order to count the number of distinct forms (for the diagrams C" H2n+1), we must in each of the above ramifications consider what is the number of distinct classes into which the points group themselves, or, say, the number of "allotrious" points. For instance, in the ramification n=3 there are two classes only; viz. a point is either terminal or medial; or, say, the number of allotrious points i =2: this is shown in the diagrams by means of the asterisks so that in each case the points which may be considered allo trious are represented by asterisks, and the number of asterisk is equal to the number of allotrious points. Thus, number of univalent diagrams C" H2n+1;. |