5 
But now that we have found that the series answering to a normal 
additive transmutation has necessarily the property of representing 
a complete transmutation, our theorem can be expressed in the 
following form, in which it is more really a ‘theorem of Mac Laurin” 
for the functional calculus, the unsatisfying point mentioned above 
having disappeared : 
A normal additive transmutation can always, in its N. F. O. 
(«), be expanded in the functional series of Mac Laurin (1), either 
for all functions of its F. F., or for a certain part of it, consisting 
of the functions belonging to a circle (3) > (0), (6) being the N.F. F.; 
and the radius 8 is never greater than o + 2a. 
