ON THE RECENT PROGRESS OF ANALYSIS. 39 
‘Comparing this with the general expression of dx, we perceive that 
Peay. ol B=&e.=0; 
and as 1 = =, (vide ante, p- 38.),* 
oy V1 — x? 
ee a 
yam Qn+a 
so that ap Rin UB 
Qx+-a 
Let 2, and x, be the two roots of our equation, we have thus to find the 
value of 
; I 1 ue Q(r+%ta) _ 
=e ala iFoe a Bayt | oid rah a) (a+ a) 
since t+ 2,=— 4. 
Hence y,da,t+ Yd t= 0, 
and v2.4 r= ec 
Since 2+ %=—a, 
and 
teas 
ae, g (a1), 
we see that af+af=1, orra=V1—2,. 
Hence, as W x=sin-'z, our result is merely this, that the sum of two ares 
is constant if the sine of one is equal to the cosine of the other. 
An infinity of analogous results may be obtained either by varying the 
form of y (e.g. by making y = “1—2®), or by changing the equation (1.). 
A formula applicable to all forms of y, and which, for each, includes all the 
results which can be established with respect to it, is, it will readily be ac- 
knowledged, one of the most general in the whole range of analysis. Abel’s 
principal result is a formula of this nature; he developed at considerable 
length the various consequences which may be deduced from it, 
Generally speaking, the number of independent variables a, 6,...¢ will 
be less than that of the different roots, 2,,..- 2; hence a certain number, 
say m, of the roots may be looked on as independent (viz, as many as there 
are quantities a, b,...c), and the rest will be functions of these. It may 
be shown that it will always be possible to make the difference py, — m con- 
stant, so that the sum of any number of the transcendents p is expressible by 
a fixed number of them, together with an algebraical and logarithmic func- 
tion of the arguments, i.e. of #,,...@m. In the case of elliptic integrals, it 
had long been known that the sum of two may be thus expressed by a third ; 
and Legendre pointed out that the sum of any number may similarly be ex- 
pressed by means of one. Accordingly it appears from the general theory, 
that in this ease 44 — m may be made equal to unity. 
9. The history of this important theory is curious. It was developed by 
Abel in an essay which he presented to the Institute in the autumn of 1826, 
when he had scarcely completed his twenty-fourth year. 
Tn a letter to M. Holmboe, appended to the edition of his collected works, 
Abel writes, “ Je viens de finir un grand traité sur une certaine classe de 
__ fonctions transcendantes pour le présenter a l'Institut, ce qui aura lieu lundi 
* The ambiguous sign of the radical is to our purpose immaterial. 
