86 
where x, y, z, X are functions of a, /3, y, has for its unique solution 
the stereographic or inversion transformation. 
The following rough analysis will give some idea of the territory 
covered by his more elaborate memoirs : — 
One of the earliest subjects that engaged his attention was 
Generalised Differentiation (“Diffehentielles a Indices Quelconques”). 
The subject is developed at considerable length in five memoirs 
printed in the 13th and 15th volumes of the Journal de Vficole 
Polytechnique (1832 --37). 
Some of his most important work relates to the Integral Calculus, 
more particularly that part of it which deals with the theory of 
elliptic and other transcendental functions. 
The earliest memoirs on the subject are two in the Journal de 
Vfcole Polytechnique (xiv. Cah. 1833, see also Comptes Rendus, 
1837), “On the Determination of Integrals whose value is 
Algebraical.” He here follows up the researches of Abel on the 
same subject ; and arrives, inter alia , at the following important 
results : — 
1. If x be any rational function of x, then fdx^-v, if algebraically 
expressible at all, can be expressed in the form P ^/x, P being 
rational. And, farther, that the integral fdx can always be 
reduced to the form 0/ ? ^/T, where T is a known rational integral 
function, and 6 a rational integral function whose coefficients have 
to be determined. This theorem enables us at once to find the value 
* 
of the integral, if it is algebraically expressible ; or else to show that 
it has no finite algebraical value. 
2. If y be an algebraical function of x, i.e., connected with x by 
means of an equation F(a?, y) = 0, which is rational and integral in 
both x and y , then, if the integral fydx is expressible explicitly 
in finite terms by means of algebraic, exponential, or logarithmic 
functions, it will be expressible in the form 
fydx = DA log u + E log v +....+ C log w ; 
where AB . . . . C are constants, and t, u, v, ... . w algebraic 
functions of x. 
Among the other memoirs on the present subject may be 
mentioned the following : — 
“ On the Elliptic Transcendents of the First and Second Species, 
