74 REPORT—1846. 
(Comptes Rendus, January 1846) pointed out the error into which he had 
fallen. 
27. In the Transactions of the Royal Irish Academy (ix. p. 151), Dr. 
Brinkley gave a geometrical demonstration of Fagnani’s theorem with respect 
to elliptic ares, and in the sixteenth volume of the same Transactions (p. 76), 
we find Landen’s theorem proved geometrically by Professor MacCullagh. 
M. Chasles has considered the subject of the comparison of elliptic ares 
by geometrical methods, and with great success. His fundamental propo- 
sition may be said to be, that if from any two points of an ellipse we draw 
two pairs of tangents to any confocal ellipse, the difference of the two arcs of 
the latter respectively intercepted by each pair of tangents is rectifiable. 
Or, what in effect is the same thing, if we fasten a string at two points in 
the circumference of an ellipse, and suppose a ring to move along the string, 
keeping it stretched, and winding it on and off the are which lies between its 
two extremities, the ring will trace out a portion of an ellipse confocal to the 
former. If for the first ellipse we substitute an hyperbola confocal with the 
second, the swm of the ares will be constant. From hence a series of theo- 
rems is deduced, remarkable not only for their elegance, but also for the 
facility with which they are obtained. They furnish constructions for the 
addition and multiplication of elliptic integrals. The whole of this investi- 
gation, of which an account is given in the ‘Comptes Rendus’ (vol. xvii. 
p- 838, and vol. xix. p. 1239), shows, like others of M. Chasles’s, how much 
is lost in treating geometrical questions by an exclusive adherence to what 
may be called the method of co-ordination. Invaluable as this method is, 
it yet often introduces considerations foreign to the problem to which it is 
applied *. 
III. 
28. The first outline of a detailed theory of the higher transcendents was 
given by Legendre in the third supplement to his ‘ Traité des Fonctions 
Elliptiques.’ He proposes to classify the transcendents comprised in the 
general formula 
f(a)de 
(x—a) Vou 
according to the degree of the polynomial ¢ 2, the first class being that in 
which the index of this degree is three or four; the second that in which it 
is five or six, and so on. The first class therefore consists of elliptic inte- 
grals; all the others may be designated as wlira-elliptic. This epithet, how- 
ever, which was proposed by Legendre, has not been so generally used as 
hyper-elliptic, which was, I believe, first used by M. Jacobi. M. Jacobi, how- 
ever, has proposed to call the higher transcendents Abelian integrals. 
The principle of Legendre’s classification is to be found in the minimum 
number of integrals to which the sum of any number of them can be reduced. 
As we know, this number is unity in the case of elliptic integrals, and by 
Abel's theorem we find that it is two in the first class of the higher trans- 
cendents, three in the next, and so on. 
Following the analogy of elliptic integrals, Legendre proposed to recognise 
three canonical forms in each class of hyper-elliptic integrals, and thus to 
divide it into three orders. The sum of any number of functions of the first 
* M. Chasles has also considered the subject of spherical conics, as well as that of the 
lines of curvature and shortest lines on an ellipsoid. The latter has recently engaged the 
attention of several distinguished mathematicians—MM. Jacobi, Joachimsthal, Liouville, 
MacCullagh and M. Roberts may be particularly mentioned. 
