the calculus of functions. 249 
but the index of the order of a function may be a fraction* or 
even a variable quantity, and such equations as the following 
might occur d ly \ x 
Vd7 dtp 
To notice the extreme difficulty of the enquiries to which 
such equations would lead, might seem superfluous, though it 
may not be deemed equally so to support my own opinion 
of their utility by the authority of one well acquainted with 
these subjects, Lacroix, in the third volume of his Traite du 
Calcul, Diff. et Int. speaking of fractional indices of differen- 
tiation, observes, C£ U Analyse offre une foule depressions 
de ce genre, qui tiennent presque toutes aux theories les 
plus importantes et les plus delicates, et les reflexions que 
j’ai exposees dans le No. 96^, me portent a croire que leur 
consideration peut contribuer beaucoup aux progres de la 
science du calcul.” 
Problem XL. 
Given the equation 
d\(x. By) d y (ax, y ) 
dx dy 
also ax = x and ( 3 *y = y. 
Put ax for x, and ( 3 y Tory, then the equation becomes 
dy (ctx, y ) dy (x, (3y) 
dux d(2y 
, dy(MX,y)l da.x d dy(x,(3y ) / dfiy \ 1 
hellce —*-(-*]= —jy—iwl 
differentiate this equation relative to y, and the original one 
relative to sci then the two results are 
~»I 
d z y (ux, y) _ da,x d. r dy (x, fiy) fd@y\ } 
dxdy dx dy \ dy \ dy j 3 
* The difficulties which occur in treating functions with negative indices are simi- , 
lar to those in which they are positive; it may however be observed, that from the 
notation we have established, the following consequences follow : 
T 0 ’ 1 (x,y) — x and T 1 ’ 0 ^, y) =y 
and generally y°> n (x, y) = x and y n>0 (x, y) = y 
also y°>° (x, x ) — X and if y l > 1 (x, y) — v, then we have 
x — J/’ 1 (v, y) and also r: y ( x , v ) 
